# The distinguishing index of graphs with infinite minimum degree

Marcin Stawiski  AGH University of Science and Technology
Faculty of Applied Mathematics,
al. Mickiewicza 30, 30-059 Krakow, Poland  Trevor M. Wilson  Miami University
Department of Mathematics
123 Bachelor Hall
301 S. Patterson Ave.
Oxford, Ohio 45056, USA

###### Abstract

The distinguishing index $D^{\prime}(G)$ of a graph $G$ is the least number of
colors necessary to obtain an edge coloring of $G$ that is preserved only by
the trivial automorphism. We show that if $G$ is a connected $\alpha$-regular
graph for some infinite cardinal $\alpha$ then $D^{\prime}(G)\leq 2$, proving
a conjecture of Lehner, Pilśniak, and Stawiski. We also show that if $G$ is a
graph with infinite minimum degree and at most $2^{\alpha}$ vertices of degree
$\alpha$ for every infinite cardinal $\alpha$, then $D^{\prime}(G)\leq 3$. In
particular, $D^{\prime}(G)\leq 3$ if $G$ has infinite minimum degree and order
at most $2^{\aleph_{0}}$.

In this note we consider undirected graphs without loops or multiple edges. An
_edge coloring_ of a graph is a function from its set of edges to a set of
elements called colors. (We do not require edge colorings to be proper.) We
say that an automorphism of a graph $G$ _preserves_ an edge coloring of $G$ if
it maps every edge to an edge of the same color. The _distinguishing index_
$D^{\prime}(G)$ of a graph $G$ is the least cardinal number $d$ such that $G$
has an edge coloring with $d$ colors that is preserved only by the trivial
automorphism. The distinguishing index is defined only for graphs with no
isolated edges and at most one isolated vertex; all graphs that we consider
will satisfy this property.

Because $D^{\prime}(G)=1$ for $G\neq K_{1}$ simply means that $G$ is
asymmetric (lacks nontrivial automorphisms), the main question about the
distinguishing index of any given graph $G$ is whether $D^{\prime}(G)\leq 2$,
in other words whether the edges of $G$ can be colored with two colors in a
way that “breaks” every nontrivial automorphism. Note that this problem is
related (but not equivalent) to the problem of the existence of an asymmetric
spanning subgraph which may be interesting on its own.

The distinguishing index was introduced by Kalinowski and Pilśniak [9]; the
corresponding notion for vertex colorings called the _distinguishing number_
$D(G)$ had been previously introduced by Babai [3] in 1977 under the name
_asymmetric colorings_. Distinguishing colorings play an important role in his
study of the graph isomorphism problem which includes the proof of the
existence of a quasi-polynomial algorithm for this problem (see [4]).
Albertson and Collins [1] reintroduced the concept of asymmetric colorings in
1996 and coined the term _distinguishing coloring_.

In this note we will consider the distinguishing index of graphs with infinite
minimum degree, possibly uncountable. Although most previous work on the
distinguishing index has focused on locally finite graphs, graphs with
infinite degrees have also been considered in [3, 5, 6, 11, 12, 13]. Note that
the very first result on the distinguishing colorings is the theorem of Babai
[3] who showed that every connected tree of infinite degree has a asymmetric
spanning forest. In this paper we generalize this result to every connected
regular graph of infinite degree.

Broere and Pilśniak [5] showed that $D^{\prime}(G)\leq 2$ for every connected
$\aleph_{0}$-regular graph $G$ and more generally for every countable graph
with what they called “good degrees”. A similar result for much larger graphs
was obtained by Lehner, Pilśniak, and Stawiski [12, Theorem 5], who showed
that $D^{\prime}(G)\leq 2$ for every connected $\alpha$-regular graph $G$
where $\alpha$ is a fixed point of the aleph sequence of cardinals, meaning
$\alpha=\aleph_{\alpha}$.111The aleph sequence is the increasing enumeration
of all infinite cardinals, beginning with $\aleph_{0}=|\mathbb{N}|$. Further
details can be found in a set theory reference such as Jech [8, Chapter 3].
They conjectured that $D^{\prime}(G)\leq 2$ holds more generally if $G$ is a
connected $\alpha$-regular graph for any infinite cardinal $\alpha$
whatsoever. We will prove this conjecture as Theorem 2 below. Our method is
necessarily quite different from the previous special cases, which relied on
the existence of $\alpha$ many cardinals less than $\alpha$. We will also
show, in Theorem 4, that $D^{\prime}(G)\leq 3$ for graphs of infinite minimum
degree that are not necessarily regular but satisfy a certain condition on the
number of vertices of each degree. Note that a version of Theorem 2 for (both
finite and infinite) connected locally finite graphs of order at least 6 was
recently proved by Kwaśny and Stawiski [10]. Combining this result with
Theorem 2 we obtained that the distinguishing index of a connected regular
graph of order at least 6 is at most 2.

Before proceeding, we briefly review some basic terminology that is mostly
standard. The set of vertices and the set of edges of a graph $G$ will be
written as $V(G)$ and $E(G)$ respectively. The _degree_ of a vertex is its
number of neighbors (adjacent vertices) and the _minimum degree_ of a graph is
the least degree of any of its vertices. A graph is _$\alpha$ -regular_, where
$\alpha$ is a cardinal, if each of its vertices has degree $\alpha$. A
_forest_ is an acyclic graph and a _tree_ is a connected forest. A spanning
subgraph of $G$ is a subgraph containing all vertices of $G$, and a spanning
tree (or forest) is a spanning subgraph that is a tree (or forest). Since we
do not require spanning forests to be maximal, a tree may have a disconnected
spanning forest, and a forest may be spanned by a smaller forest.

The following lemma turns out to be useful for investigating the
distinguishing index of graphs of infinite minimum degree. Theorem 2 will
follow easily from a special case of it, whereas Theorem 4 will require some
additional work. To prove the lemma we will need a theorem of Babai [3],
namely that for every infinite cardinal $\alpha$ there is a family of
$2^{\alpha}$ pairwise non-isomorphic asymmetric trees of order $\alpha$.
Actually Babai proved more specifically that such a family can be constructed
in which each tree has vertices of two prescribed degrees only, such as 1 and
$\alpha$. This more specific statement will not be needed here.

######  Lemma 1.

Let $(T_{i})_{i\in I}$ be a family of trees, each of which is regular of
infinite degree, and at most $2^{\alpha}$ of which are $\alpha$-regular for
any particular infinite cardinal $\alpha$. Then the disjoint union
$\bigcup_{i\in I}T_{i}$ has an asymmetric spanning forest with no isolated
vertices.

###### Proof.

It suffices to show that for each infinite cardinal $\alpha$, the disjoint
union of all the $\alpha$-regular trees in the family has an asymmetric
spanning forest in which each component has order $\alpha$. Then we may take
the disjoint union of these asymmetric forests for all infinite cardinals
$\alpha$ to obtain the desired asymmetric forest spanning all of
$\bigcup_{i\in I}T_{i}$, since an automorphism cannot interchange components
of two different orders.

We may therefore assume without loss of generality that each tree $T_{i}$ in
the family is $\alpha$-regular for the same fixed infinite cardinal $\alpha$,
and that the index set $I$ has cardinality at most $2^{\alpha}$. Under this
assumption, our task is to show that the disjoint union $\bigcup_{i\in
I}T_{i}$ has an asymmetric spanning forest in which each component has order
$\alpha$.

There is a family of $2^{\alpha}$ pairwise non-isomorphic asymmetric trees of
order $\alpha$ by Babai [3]. (For example, there is such a family of trees
having vertices of degrees 1 and $\alpha$ only.) Fixing a set $J$ of
cardinality $\alpha$ to use as another index set, we have $|I\times J|\leq
2^{\alpha}\cdot\alpha=2^{\alpha}$, so there is a family $(A_{i,j})_{i\in
I,j\in J}$ of pairwise non-isomorphic asymmetric trees of order $\alpha$. The
disjoint union $\bigcup_{i\in I,j\in J}A_{i,j}$ of this family is an
asymmetric forest in which each component has order $\alpha$, so it suffices
to find a spanning forest of $\bigcup_{i\in I}T_{i}$ isomorphic to
$\bigcup_{i\in I,j\in J}A_{i,j}$. Treating each $i\in I$ separately, we will
find a spanning forest of $T_{i}$ isomorphic to $\bigcup_{j\in J}A_{i,j}$.

Let us consider a fixed index $i\in I$. Omitting this index from our notation,
we can describe our task more simply as follows. Given an $\alpha$-regular
tree $T$ and a family $(A_{j})_{j\in J}$ of trees of order $\alpha$, where $J$
has cardinality $\alpha$, we want to find a spanning forest of $T$ isomorphic
to $\bigcup_{j\in J}A_{j}$. (The remaining argument will not use the
assumption that the trees $A_{j}$ are pairwise non-isomorphic and asymmetric;
they can be any trees of order $\alpha$.)

To show that the forest $F=\bigcup_{j\in J}A_{j}$ is isomorphic to a spanning
forest of $T$, we will add some edges to $F$ to obtain an $\alpha$-regular
tree $T^{\prime}$, which must be isomorphic to $T$ because any two
$\alpha$-regular trees are isomorphic. We first write the index set $J$ as a
disjoint union of sets $J_{n}$, $n\in\mathbb{N}$ where $J_{0}$ is a singleton
set $\\{j_{0}\\}$ and each $J_{n+1}$ has cardinality $\alpha$. Then we can
rewrite the forest $F=\bigcup_{j\in J}A_{j}$ as the following disjoint union
of subforests: $F=\bigcup_{n\in\mathbb{N}}F_{n}$ where $F_{n}=\bigcup_{j\in
J_{n}}A_{j}$.

Note that each subforest $F_{n}$ has order $\alpha$, and that $F_{0}$ is just
a single tree (namely $A_{j_{0}}$) whereas each $F_{n+1}$ has $\alpha$
components. For each $n\in\mathbb{N}$, we may add to $F$ a set $E_{n}$ of
$\alpha$ edges between $F_{n}$ and $F_{n+1}$ in such a way that

  1. 1.

every vertex of $F_{n}$ is incident with $\alpha$ edges in $E_{n}$, and

  2. 2.

every component of $F_{n+1}$ is incident with one and only one edge in
$E_{n}$.

Let $T^{\prime}$ be the graph obtained by adding the edges in
$\bigcup_{n\in\mathbb{N}}E_{n}$ to the forest $F$. By (1), the graph
$T^{\prime}$ is $\alpha$-regular. By the “only one” part of (2) and the fact
that each $F_{n}$ is acyclic, $T^{\prime}$ is acyclic. By the “(at least) one”
part of (2) and the fact that $F_{0}=A_{j_{0}}$ is connected, $T^{\prime}$ is
connected. Therefore $T^{\prime}$ is an $\alpha$-regular tree spanned by $F$
as desired. ∎

Our applications of Lemma 1 will only use the fact that $\bigcup_{i\in
I}T_{i}$ has an asymmetric spanning subgraph with no isolated vertices, and
not the fact that this subgraph is a forest.

The lemma can be applied in the special case of a single regular tree $T_{i}$
to prove the conjecture of Lehner, Pilśniak, and Stawiski [12, Conjecture 6].
The only other ingredient we will need is a result of Andersen and Thomassen
[2, Theorem 2′], which says that for every infinite cardinal $\alpha$, every
connected $\alpha$-regular graph has an $\alpha$-regular spanning tree.

######  Theorem 2.

Let $G$ be a connected $\alpha$-regular graph where $\alpha$ is an infinite
cardinal. Then $D^{\prime}(G)\leq 2$.

###### Proof.

The graph $G$ has an $\alpha$-regular spanning tree, which in turn has an
asymmetric spanning forest $F$ by Lemma 1. Coloring the edges of $F$ red and
the other edges of $G$ blue, we obtain an edge coloring of $G$ that is
preserved only by the trivial automorphism. ∎

The following proposition shows that the hypothesis of connectedness can be
removed if we impose a certain limitation on the order, or equivalently for
$\alpha$-regular graphs, a limitation on the number of components. To see that
some such limitation is necessary, letting $\alpha$ be an infinite cardinal
and denoting the least cardinal greater than $2^{\alpha}$ by
$(2^{\alpha})^{+}$, observe that the disjoint union of $(2^{\alpha})^{+}$
copies of any graph of order $\alpha$ has distinguishing index
$(2^{\alpha})^{+}$ because any edge coloring with at most $2^{\alpha}$ colors
admits an automorphism interchanging two components.

######  Proposition 3.

Let $G$ be an $\alpha$-regular graph of order at most $2^{\alpha}$ where
$\alpha$ is an infinite cardinal. Then $D^{\prime}(G)\leq 2$.

###### Proof.

The graph $G$ has at most $2^{\alpha}$ components and we may choose an
$\alpha$-regular spanning tree for each component. Applying Lemma 1 to this
family of trees yields an asymmetric spanning forest $F$ of $G$, and as before
we may color the edges of $F$ red and the other edges of $G$ blue to obtain an
edge coloring of $G$ that is preserved only by the trivial automorphism. ∎

Note that in the proposition above we obtained even a stronger result. Namely
each $\alpha$-regular graph of order at most $2^{\alpha}$ where $\alpha$ is an
infinite cardinal has an asymmetric spanning forest. It also generalizes the
mentioned result of Babai which showed that each infinite regular tree has an
asymmetric spanning forest. We showed it for every connected regular graph of
infinite degree.

The following theorem shows that the hypothesis of regularity can also be
removed if we allow an additional color and we again impose a certain
limitation, this time on the number of vertices of each degree. To see that
some such limitation is necessary even for connected graphs, observe that the
complete bipartite graph $K_{\alpha,(2^{\alpha})^{+}}$ has distinguishing
index $(2^{\alpha})^{+}$ because any edge coloring with at most $2^{\alpha}$
colors admits an automorphism interchanging two vertices of degree $\alpha$.

To prove the theorem we will need another result of Andersen and Thomassen [2,
Theorem 2], which says that every graph $G$ of infinite minimum degree
$\delta$ contains (as a subgraph, not necessarily induced) a $\delta$-regular
tree. We will also need the fact that such a tree may be taken to include any
given vertex of $G$, which is apparent from their proof.

######  Theorem 4.

222We note that this theorem is due to second author solely.

Let $G$ be a graph with infinite minimum degree and at most $2^{\alpha}$
vertices of degree $\alpha$ for every infinite cardinal $\alpha$. Then
$D^{\prime}(G)\leq 3$.

###### Proof.

Let us call a tree $T$ contained in $G$ _degree-realizing_ if for some
infinite cardinal $\alpha$, $T$ is $\alpha$-regular and some vertex of $T$ has
degree $\alpha$ in $G$. Note that $G$ contains at least one degree-realizing
tree, because we may choose a vertex whose degree equals the minimum degree
$\delta(G)$ of $G$ and then find a $\delta(G)$-regular tree containing that
vertex.

Let $(T_{i})_{i\in I}$ be a pairwise disjoint family of degree-realizing trees
in $G$ that is maximal in the sense that $G$ contains no other degree-
realizing tree disjoint from every $T_{i}$. (Such a family exists by Zorn’s
lemma.) Note that for each infinite cardinal $\alpha$, at most $2^{\alpha}$
trees in this family are $\alpha$-regular. This is because different
$\alpha$-regular trees in the family are disjoint and degree-realizing, so
they must contain different vertices of degree $\alpha$ in $G$, and at most
$2^{\alpha}$ such vertices exist in $G$ by our hypothesis. It follows by Lemma
1 that $\bigcup_{i\in I}T_{i}$ has an asymmetric spanning forest $F$ with no
isolated vertices.

Color all edges of $F$ red and color any other edges of the induced subgraph
of $G$ on $V(F)$ blue. Because $F$ might not span $G$, we may need to color
some additional edges. For these edges, we will use the colors green and blue.
We will never color any more edges outside of $E(F)$ red and $F$ has no
isolated vertices, so any automorphism of $G$ preserving our edge coloring
will fix $V(F)$ setwise, and therefore also pointwise since $F$ is asymmetric.
This will allow us to use $V(F)$ as a kind of foundation or anchor for the
rest of our coloring.

Let $H_{0}$ be the induced subgraph of $G$ on $V(F)$ and let $c_{0}$ be the
edge coloring of $H_{0}$ that colors all edges of $F$ red and any other edges
of $H_{0}$ blue, as mentioned previously. We define a _good partial coloring_
to be a pair $(H,c)$ where

  * •

$H$ is an induced subgraph of $G$ containing $H_{0}$,

  * •

every automorphism of $G$ fixing $V(F)$ setwise also fixes $V(H)$ setwise,

  * •

$c$ is an edge coloring of $H$ extending $c_{0}$ and using only green and blue
outside $E(F)$, and

  * •

every automorphism of $G$ fixing $V(F)$ (and therefore also $V(H)$) setwise
and preserving $c$ fixes $V(H)$ pointwise.

Note that $(H_{0},c_{0})$ itself is a good partial coloring. Let $\mathcal{P}$
be the poset of all good partial colorings ordered by extension:
$(H,c)\leq(H^{\prime},c^{\prime})$ means $H^{\prime}$ contains $H$ and
$c^{\prime}\restriction E(H)=c$. It’s not hard to check that for any chain in
$\mathcal{P}$, its union (in the obvious sense) is also in $\mathcal{P}$.
Since the union is an upper bound for the chain, a maximal good partial
coloring exists by Zorn’s lemma.

Let $(H,c)$ be a maximal good partial coloring. If $H=G$, then $c$ is an edge
coloring of the entire graph and we are done: only the edges of $F$ are
colored red and $F$ has no isolated vertices, so every automorphism of $G$
preserving $c$ must fix $V(F)$ setwise (in fact pointwise since $F$ is
asymmetric) and must therefore be the identity by the definition of “good
partial coloring”.

Now we suppose that $H\neq G$ and argue toward a contradiction. Let
$G\setminus H$ be the induced subgraph of $G$ on $V(G)\setminus V(H)$, let
$\delta(G\setminus H)$ be the minimum degree of $G\setminus H$, let $X$ be the
set of all vertices of $G\setminus H$ whose degree in $G\setminus H$ equals
this minimum degree $\delta(G\setminus H)$, and let $H^{\prime}$ be the
induced subgraph of $G$ with vertex set $V(H^{\prime})=V(H)\cup X$. We will
define an edge coloring $c^{\prime}$ of $H^{\prime}$ extending $c$ such that
$(H^{\prime},c^{\prime})$ is a good partial coloring, contradicting the
maximality of $(H,c)$.

Because $(H,c)$ is good, every automorphism of $G$ fixing $V(F)$ setwise fixes
$V(H)$ setwise, and because $X$ and $H^{\prime}$ were defined from $H$ in an
invariant way, it follows that every automorphism of $G$ fixing $V(F)$ setwise
also fixes $V(H^{\prime})$ setwise. Again because $(H,c)$ is good, every
automorphism of $G$ fixing $V(F)$ (and therefore also $V(H)$) setwise and
preserving $c$ fixes $V(H)$ pointwise. Therefore it will suffice to color the
edges between $X$ and $H$ with the colors green and blue in a way to ensure
that every color-preserving automorphism of $G$ fixing $V(H)$ pointwise must
also fix $X$ pointwise. (Any edges between two vertices in $X$ will not be
needed for this symmetry breaking, so let us arbitrarily choose to color them
blue rather than green.)

Let $\sim$ be the equivalence relation on $X$ where $u\sim v$ means that $u$
and $v$ have the same set of neighbors in $H$. Letting $Y\subseteq X$ be an
equivalence class, note that every automorphism of $G$ fixing $V(H)$ pointwise
must fix $Y$ setwise. To ensure that every color-preserving automorphism of
$G$ fixing $V(H)$ pointwise must also fix $Y$ pointwise, it suffices to color
the edges between $Y$ and $H$ in such a way that any two distinct vertices in
$Y$ have different sets of green neighbors in $H$ (meaning neighbors in $H$
joined to them by green edges.) To show this is possible, we must show that
$Y$ has at most $2^{\alpha_{Y}}$ vertices, where $\alpha_{Y}$ is defined as
the number of neighbors in $H$ that each vertex in $Y$ has.

The hypothesis of the theorem guarantees that $G$ has at most $2^{\alpha_{Y}}$
vertices of degree $\alpha_{Y}$, so it will suffice to show that every vertex
$v\in Y$ has degree $\alpha_{Y}$ in $G$. If not, then since $\alpha_{Y}$ is
its number of neighbors in $H$ and all degrees in $G$ are infinite, its degree
in $G$ must equal its degree in $G\setminus H$, which is the minimum degree
$\delta(G\setminus H)$ of $G\setminus H$ since $v\in X$. Letting $T$ be a
$\delta(G\setminus H)$-regular tree in $G\setminus H$ containing $v$, which
exists by the theorem of Andersen and Thomassen, this tree $T$ therefore
counts as degree-realizing in $G$. However, this contradicts the maximality of
the pairwise disjoint family $(T_{i})_{i\in I}$ because $T$ is contained in
$G\setminus H$ whereas each $T_{i}$ is contained in $F$ and therefore in $H$.
This contradiction proves the theorem. ∎

Theorem 4 may leave some room for improvement. Although we could look for ways
to weaken the hypothesis on $G$, the most obvious question is whether or not
three colors are actually needed:

######  Question 5.

If $G$ is a graph with infinite minimum degree and at most $2^{\alpha}$
vertices of degree $\alpha$ for every infinite cardinal $\alpha$, must
$D^{\prime}(G)\leq 2$?

Note that a positive answer to this question cannot be obtained by finding an
asymmetric spanning forest of $G$ as in the proofs of Theorem 2 and
Proposition 3. This is because for any two infinite cardinals $\alpha<\beta$,
the complete bipartite graph $K_{\alpha,\beta}$ (which satisfies the
hypothesis of Theorem 4 if $\beta\leq 2^{\alpha}$) has no asymmetric spanning
forest. As remarked by Dirac and Thomassen [7, p. 409], any spanning forest of
$K_{\alpha,\beta}$ for infinite cardinals $\alpha<\beta$ must have either
$\beta$ isolated vertices or $\beta$ leaves ($\beta$ of which must share a
common neighbor) and in either case interchanging two of these vertices
produces a nontrivial automorphism of the spanning forest.

A special case of Theorem 4 and Question 5 may be worth noting separately:

######  Corollary 6.

If $G$ be a graph with infinite minimum degree and order at most
$2^{\aleph_{0}}$, then $D^{\prime}(G)\leq 3$.

######  Question 7.

If $G$ is a graph with infinite minimum degree and order at most
$2^{\aleph_{0}}$, must $D^{\prime}(G)\leq 2$?

## Acknowledgement

The authors would like to thank Louis DeBiasio for his helpful comments on a
draft of the article.

## References

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††thanks: Authors to whom correspondence should be addressed:
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>Authors to whom
correspondence should be addressed<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
# Non-invasive time-sorting in radio-frequency compressed ultrafast electron
diffraction

Lingrong Zhao  Jun Wu  Zhe Wang  Heng Tang  Xiao Zou  Tao Jiang  Pengfei Zhu
Key Laboratory for Laser Plasmas (Ministry of Education), School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong
University, Shanghai 200240, China  Dao Xiang  Key Laboratory for Laser
Plasmas (Ministry of Education), School of Physics and Astronomy, Shanghai
Jiao Tong University, Shanghai 200240, China  Collaborative Innovation Center
of IFSA (CICIFSA), Shanghai Jiao Tong University, Shanghai 200240, China
Tsung-Dao Lee Institute, Shanghai 200240, China  Zhangjiang Institute for
Advanced Study, Shanghai Jiao Tong University, Shanghai 200240, China  Jie
Zhang  Key Laboratory for Laser Plasmas (Ministry of Education), School of
Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
Collaborative Innovation Center of IFSA (CICIFSA), Shanghai Jiao Tong
University, Shanghai 200240, China

###### Abstract

We demonstrate a non-invasive time-sorting method for ultrafast electron
diffraction (UED) experiments with radio-frequency (rf) compressed electron
beams. We show that electron beam energy and arrival time at the sample after
rf compression are strongly correlated such that the arrival time jitter may
be corrected through measurement of the beam energy. The method requires
minimal change to the infrastructure of most of the UED machines and is
applicable to both keV and MeV UED. In our experiment with $\sim$3 MeV beam,
the timing jitter after rf compression is corrected with 35 fs root-mean-
square (rms) accuracy, limited by the $3\times 10^{-4}$ energy stability. For
keV UED with high energy stability, sub-10 fs accuracy in time-sorting should
be readily achievable. This time-sorting technique allows us to retrieve the
2.5 THz oscillation related to coherent A1g phonon in laser excited Bismuth
film and extends the temporal resolution of UED to a regime far beyond the
100-200 fs rms jitter limitation.

††preprint: AIP/123-QED

##  I INTRODUCTION

Ultrafast electron diffraction (UED) has emerged as a powerful tool with high
temporal-spatial resolving power, providing direct insight into the structural
dynamics of matter (Chergui and Zewail, 2009; Sciaini and Miller, 2011). In
UED experiments, the dynamics are initiated by an ultrashort pump laser and
probed by a delayed electron pulse. By recording diffraction patterns at a
series of pump-probe delays, it is possible to retrieve atomic changes
following laser excitation. This approach has been applied to study dynamics
in phase transition Siwick  _et al._ (2003); Baum, Yang, and Zewail (2007);
Gedik  _et al._ (2007); Ernstorfer  _et al._ (2009); Sciaini  _et al._ (2009);
Eichberger  _et al._ (2010); Haupt  _et al._ (2016); Mo  _et al._ (2018), to
reveal transient states Raman  _et al._ (2008); Morrison  _et al._ (2014);
Kogar  _et al._ (2020); Zhou  _et al._ (2021); Duan  _et al._ (2021), and to
visualize molecular dynamics Ihee  _et al._ (2001); Yang  _et al._ (2016,
2018); Wolf  _et al._ (2019). With the advent of femtosecond laser, the time
resolution of UED is primarily limited by the pulse width and timing jitter of
the electron bunch. Coulomb repulsion is the main effect that accounts for
lengthening of electron pulse width as electron beam propagates from the
source to the sample. To circumvent this limitation, many efforts have been
made in the past two decades, e.g. reducing the propagation distance Siwick
_et al._ (2003); Baum, Yang, and Zewail (2007), reducing the bunch charge to
single electron Fill  _et al._ (2006); Veisz  _et al._ (2007); Aidelsburger
_et al._ (2010), increasing the electron energy to relativistic regime
Hastings  _et al._ (2006); Li  _et al._ (2009); Musumeci  _et al._ (2010);
Murooka  _et al._ (2011); Fu  _et al._ (2014); Zhu  _et al._ (2015);
Weathersby  _et al._ (2015); Manz  _et al._ (2015), and compressing the beam
with rf buncherVan Oudheusden  _et al._ (2007, 2010); Gao  _et al._ (2012);
Chatelain  _et al._ (2012, 2014); Maxson  _et al._ (2017); Zandi  _et al._
(2017); Zhao  _et al._ (2018), THz buncherKealhofer  _et al._ (2016); Zhao
_et al._ (2020); Snively  _et al._ (2020) and double bend achromat Qi  _et
al._ (2020); Kim  _et al._ (2020).

The most widely used bunch compression method for UED is velocity bunching
with an rf buncher Van Oudheusden  _et al._ (2007, 2010); Gao  _et al._
(2012); Chatelain  _et al._ (2012, 2014); Maxson  _et al._ (2017); Zandi  _et
al._ (2017); Zhao  _et al._ (2018), and the shortest bunch duration is below
10 fs rms with bunch charge on the order of 10 fC for MeV beams Maxson  _et
al._ (2017); Zhao  _et al._ (2018). However, previous experiments have shown
that with rf buncher, the electron beam pulse width is reduced at the cost of
increasing the timing jitter which is typically measured to be about 100-200
fs rms Chatelain  _et al._ (2012); Gao  _et al._ (2012); Zandi  _et al._
(2017); Zhao  _et al._ (2018). The primary cause for this timing jitter is the
phase jitter between the laser oscillator and rf electronics. Specifically,
the phase jitter in the rf cavity leads to beam energy jitter which is further
converted into timing jitter at the sample after passing through a drift with
longitudinal dispersion. While the timing jitter can be measured with a THz
deflector Zhao  _et al._ (2018); Li  _et al._ (2019); Zhao  _et al._ (2019),
and very recently a THz-streaking based method Othman  _et al._ (2021) has
been used to correct the jitter, the method is invasive and it nonetheless
introduces changes to the diffraction pattern which may be difficult to
differentiate from those by laser excitation.

In this paper, we demonstrate a non-invasive time-sorting method to record the
arrival time of rf compressed relativistic electron beam through measurement
of the beam energy of the un-diffracted beam. Our measurements show that after
rf compression the beam arrival time at the sample is strongly correlated with
beam energy. Because the detector has a central hole to allow the un-
diffracted beam to path through, it is straightforward to simultaneously
measure both the diffraction pattern and beam arrival time with a downstream
energy spectrometer. This method has been used to retrieve the A1g phonon
oscillation in laser excited Bismuth. In contrast, no oscillation has been
observed in the raw data without jitter correction. This measure-and-sort
method is applicable to both keV and MeV UED and can be used to correct both
short-term timing jitter and long-term timing drift in rf-compressed UED. In
our experiment, the timing jitter of the rf compressed beam is corrected with
35 fs rms accuracy, limited by the energy stability of the electron beam
before the buncher. Since keV UED has high stability in beam energy, an
accuracy of a few femtosecond should be readily achievable with this time-
sorting method.

##  II Principle of bunch compression and source of timing jitter

In this section, we briefly discuss the physics behind the time-sorting
technique. For simplicity, we consider a MeV electron beam with negligible
energy chirp (correlation between an electron’s energy and longitudinal
position). In the most common velocity bunching scheme, the beam is sent
through an rf buncher cavity at the zero-crossing phase. Because the bunch
head is decelerated and the bunch tail is accelerated, the beam ends up with a
negative energy chirp (bunch head has lower energy than bunch tail) at the
exit of the buncher. The energy chirp can be calculated as,

$h=\dfrac{d\delta}{dz}=-\dfrac{2\pi V}{E\lambda}$ (1)

where $E$ is the beam energy, $V$ and $\lambda$ are the voltage and wavelength
of the rf buncher, respectively. The longitudinal dispersion of a drift with
length $L$ is $R_{56}=cdt/d\delta\approx-L/\gamma^{2}$, where $\gamma$ is the
Lorentz factor of the electron. By matching the longitudinal dispersion with
the beam energy chirp such that $hR_{56}=1$, the beam will be fully compressed
after a drift. In this case, the electrons with higher energy at the bunch
tail exactly catch up with the lower energy electrons at the bunch head after
the drift.

Ideally, the electron beam passes through the buncher at zero-crossing phase
such that the energy chirp is imprinted without changing the beam centroid
energy. In realistic cases, the phase jitter of the rf field leads to change
of the beam centroid energy. Taken the phase seen by the electron beam to be
$\Delta\phi$ (much smaller than 1), at the buncher exit the change of beam
centroid energy is $\Delta\delta\approx(V/E)\Delta\phi$. After the drift, this
energy change will result in a time-of-flight change of $\Delta t=\Delta\delta
R_{56}/c$. Under the full compression condition, the time-of-flight change is
simplified to $\Delta t=-\Delta\phi_{t}$, where
$\Delta\phi_{t}=\Delta\phi/(kc)$ is the normalized phase jitter that has the
same unit as time and $k$ is the wave number of the rf and $c$ is speed of
light. From discussions above, one can see that the timing jitter is
correlated with the energy jitter with the coefficient being the longitudinal
dispersion of the drift. Therefore, the timing jitter may be determined by
measuring the beam energy after rf compression.

Simulation with General Particle Tracer code GPT  is used to verify this
dependence and to also take into account the effect of rf amplitude and phase
jitter in both the photocathode rf gun and buncher cavity. Following our
experimental setup, in this simulation the electron beam is produced in a 1.6
cell s-band (2856 MHz) photocathode rf gun and compressed in a 5 cell c-band
(5712 MHz) buncher. The accelerating gradient and launching phase in the gun
are set to 58 MV/m and 38 degrees, respectively. The phase is chosen to
minimize the timing jitter at the entrance to the buncher Li and Tang (2009).
The buncher is located at 0.7 m downstream of the gun and the bunching voltage
is set to 0.9 MV which resulted in full compression after a drift of 1.0 m.
The amplitude jitter and phase jitter in the two rf cavities are both set to
0.03% and 150 fs, respectively. In the simulation, the kinetic energy of the
electron beam is 2.56 MeV and the corresponding $R_{56}$ from the exit of the
rf buncher to the sample is -2.77 cm (or -92.3 ps).

10000 runs of simulation with single electron are performed and the electron’s
arrival time and energy at the sample are shown in Fig. 1(a) (blue dots) where
one can see that the arrival time of the electrons is linearly correlated with
their energy. The coefficient is equal to the longitudinal dispersion of the
drift. The rms timing jitter is about 160 fs rms, as shown in Fig. 1(b). After
removing the linear term, the residual timing jitter (magenta dots) which can
not be corrected by measuring the beam energy is about 30 fs rms. Analysis
shows that the residual jitter is comparable to the beam timing jitter at the
entrance to the rf buncher which strongly depends on rf amplitude jitter in
the gun. The simulation was repeated by reducing the rf amplitude jitter in
the gun to 0.01% while keeping other parameters unchanged, and the simulation
results are shown in Fig. 1(c) and Fig. 1(d) where one can see that the
residual jitter is similarly reduced by 3 times to about 10 fs rms. In keV UED
with a DC electron gun, relative ripple of the accelerating voltage is
typically less than $10^{-5}$ and the beam arrival time should be determined
with an accuracy well below 10 fs using this method.

Figure 1:  Simulated electron timing jitter and energy jitter at the sample
with the rf amplitude jitter set to 0.03% (a) and 0.01% (c). Panel (b) and (d)
are the corresponding distributions of the timing jitter in (a) and (c)
respectively. Figure 2: (a) Schematic of the experimental setup. The 800 nm
laser is split into three pulses with one pulse for producing electron beam,
the second pulse to excite dynamics in the sample, and the third pulse for
producing THz radiation. The electron beam with about 100 fC charge is
compressed with an rf buncher. The pulse duration and time jitter is measured
with THz streaking in a dielectric-lined waveguide. The diffraction pattern is
measured with screen P1 which has a hole to allow the un-diffracted beam to
pass through. The distance between the sample and screen P1 is about 1.8 m. A
dipole magnet downstream of P1 is used to measure the energy of the un-
diffracted beam at screen P2. BS for beam splitter.

##  III Electron beam compression and characterization

The schematic of our experiment to demonstrate non-invasive time-sorting is
shown in Fig. 2. For producing electron bunches with sufficient charge to
generate high quality single shot diffraction patterns, a set of $\alpha$BBO
crystals are used to shape the temporal profile of the UV pulse from about 50
fs Gaussian distribution to $\sim$2 ps FWHM flat-top distribution. The
$\sim$2.6 MeV electron beam is compressed by a C-band (5712 MHz) rf buncher
cavity. The pulse duration and arrival time jitter of the electron beam at the
sample is characterized by a THz deflector Zhao  _et al._ (2019).

The THz pulse is generated through optical rectification in LiNbO3 crystal
Hebling  _et al._ (2002). An off axis parabolic (OAP) mirror collects and
focuses the THz radiation with vertical polarization into a dielectric-lined
waveguide (DLW) where HEM11 mode is excited Gallot  _et al._ (2000). The
electron beam in the THz deflector receives a vertical time-dependent angular
streaking, which maps its time information into spatial distribution on screen
P1.

We first measured the streaking deflectogram with a short electron beam
($\sim$100 fs rms) by removing the BBO crystals. After both the spatial and
temporal overlap between the THz and electron beam are optimized, the beam
deflectogram [Fig. 3(a)] is measured with the timing of the THz beam varied in
30 fs step. The maximal streaking ramp (around t = 2.4 ps region in Fig. 3(a))
is found to be about 6.0 $\mu$rad/fs. The dynamic range of this measurement
where the rate of angular change is approximately linear is about 400 fs. The
accuracy of the arrival time measurement is mainly affected by the fluctuation
of the centroid divergence of the electron beam, leading to temporal offset in
the measurement. In this experiment, the beam centroid fluctuation on P1 with
THz off is measured to be about 9.0 $\mu$rad, corresponding to an accuracy of
about 1.5 fs for the jitter determination. The transverse beam size at screen
P1 was measured to be about 190 $\mu$m with the THz off [Fig. 3(b)], and the
temporal resolution in beam temporal profile measurement is estimated to be
about 18 fs.

After the streaking deflectogram is obtained, the BBO crystals are then
inserted back to produce a 2-ps flat-top UV pulse and in this case an electron
bunch with similar pulse width is produced. With the rf buncher off, the
streaked beam has a double-horn distribution [Fig. 3(c)], indicating that the
electron pulse extends over at least half of the period of the THz streaking
field. We then turned on the C-band buncher for compressing the beam. The
buncher voltage is varied until the smallest streaked beam size on screen P1
is obtained. In this case, the beam reaches shortest bunch length at the
sample and the streaked beam is shown in Fig. 3(d).

The bunch length at full compression can be estimated by an analysis of the
vertical projections with and without THz streaking. The red circles and blue
dash curve in Fig. 3(e) are the profiles of the compressed beam with and
without THz streaking, respectively. By fitting the raw streaked distribution
with a Gaussian function (black), the bunch length is estimated to be about 30
fs rms. After subtracting the contribution from the intrinsic beam size, the
bunch length after deconvolution is estimated to be 25 fs rms.

Figure 3:  (a) Streaking deflectogram with maximal streaking rate of 6.0
$\mu$rad/fs. (b) Raw distribution of the electron beam with the THz off. Panel
(c) and (d) are distributions of streaked beam before (c) and after (d) rf
compression. (e) Fit to the experimental trace to estimate the bunch length at
full compression.

##  IV Timing jitter measurement

Because of the 100-200 fs rf phase jitter, the electron beam is compressed at
the cost of increasing the timing jitter to a similar level. Under full
compression condition, 50 consecutive measurements of THz-streaked beam
profiles are shown in Fig. 4(a). For comparison, we also included consecutive
measurements of the beam profile with THz off (the first 10 shots). The
arrival time of the electron beam at the sample is determined from the
centroid of the streaked profile. The timing jitter at full compression
collected over 2500 shots is estimated to be about 140 fs (rms), as shown in
Fig. 4(b).

Since the path length from the buncher to the sample is fixed, the change in
time-of-flight is essentially related to the change of velocity which strongly
depends on beam energy. In order to show the correlation between the beam
energy and arrival time and estimate the precision of the proposed time-
sorting method, we conducted a separate measurement at screen P2 downstream of
the energy spectrometer. Because the electron beam is bent in horizontal
direction by the energy spectrometer and streaked in vertical direction by the
THz deflector, the horizontal axis on screen P2 becomes the energy axis and
the vertical axis becomes the time axis. This allows us to measure both the
arrival time and beam energy simultaneously. The measured correlation between
the arrival time and centroid energy of the beam is shown in Fig. 4(c) (blue
dots), where one can see that the beam timing jitter is indeed linearly
correlated with the beam energy jitter, i.e., $\Delta$t = R$\times\Delta$E/E,
with $R$ determined to be about -93 ps, in good agreement with the value of
the longitudinal dispersion of the drift. After removing the linear term, the
residual timing jitter that limits the accuracy of jitter correction is about
35 fs rms as shown in Fig. 4(d). With this jitter correction, the temporal
resolution in this rf-compressed UED has been improved from about 150 fs to
about 50 fs rms, comparable to that achieved in a state-of-the-art double bend
achromat based UED Qi  _et al._ (2020); Kim  _et al._ (2020), yet with greatly
simplified infrastructure. The accuracy in jitter correction is mainly limited
by the rf amplitude stability of the photocathode rf gun and sub-10 fs
correction accuracy should be readily achievable for keV UED with greatly
improved beam energy stability.

Figure 4:  (a) Consecutive measurement of beam arrival time with THz
streaking. (b) Distribution of the electron beam arrival time collected over
2500 shots. A Gaussian fit to the distribution yields a timing jitter of about
140 fs (rms) between the electron beam and THz pulse. (c) Correlation of the
beam energy and arrival time. (d) Distribution of the residual jitter after
removing the linear term.

##  V Non-invasive jitter correction in UED

Figure 5:  (a) Single shot diffraction pattern of the single crystal
Bismuth.(b) Measured evolution of the Bragg reflection highlighted in (a) by
white dashed circle with and without correction of the time jitter. (c) 20
selected data points at 0.5 ps time delay before timing correction (blue
triangle) and the corresponding distribution after timing correction (red
circle).

The feasibility of this non-invasive time-sorting technique is demonstrated in
a pump-probe experiment to measure the A1g phonon dynamics of Bismuth excited
by a 800 nm laser. For pump-probe experiment, after the full compression
condition is achieved at the sample, the OAP and DLW are removed from the beam
path with an in-vacuum stage and the sample is inserted into the beam path. A
pump laser pulse is used to excite the ultrafast structural dynamics in the
sample. The diffraction pattern and beam energy of the unscattered beam is
synchronously recorded on screen P1 and P2, respectively. It should be noted
that screen P1 has a 2-mm-diameter hole in the center to let the un-diffracted
beam to pass through.

A 30-nm-thick (110) oriented Bismuth film is grown by molecular beam epitaxy
on KCl(100) substrate Moriena  _et al._ (2012). With femtosecond
photoexcitation of charge carriers, coherent A1g optical phonon, corresponding
to atomic vibrations parallel to the trigonal axis of the rhombohedral unit
cell of Bismuth is produced Cheng  _et al._ (1990). The atomic vibration leads
to oscillation in diffraction intensity which has been widely used to test the
temporal resolution of the instrument Beaud  _et al._ (2007); Harmand  _et
al._ (2013). To measure this structural change the sample is rotated by
approximately 37 degrees with respect to normal incidence. Representative
single shot diffraction pattern obtained with our rf compressed beam is shown
in Fig. 5(a). The Bragg reflections sensitive to A1g phonon are marked by
white dashed circles. A 800 nm, 25 fs FWHM laser with a fluence of about 1.5
mJ/cm2 is used to excite the dynamics. The time delay of the pump laser and
electron beam is controlled by a delay stage with step size of 40 fs.
Diffraction pattern and energy of the unscattered electron beam are collected
synchronously at 10 Hz, limited by the frame rate of our CCD camera. In this
experiment, 800 diffraction patterns (20 diffraction patterns for each time
step) are collected and the measured intensity evolution of the highlighted
Bragg reflection before and after correcting the time jitter are shown in Fig.
5(b). The A1g coherent phonon oscillation at about 2.5 THz is clearly resolved
after jitter correction while no oscillation is observed for the raw data
without jitter correction. Fig. 5(c) shows the process of jitter correction
for the 20 data points collected at time delay of 0.5 ps. Before correction
(blue triangle), the data points have different intensity, but are assigned
with the same delay time. After correcting jitter based on the measured beam
energy for each data, the corrected data points (red circle) have different
delay time (diffaction intensity for each data is not changed in the process)
and now the data has a better match to the A1g coherent phonon oscillation.
Similar procedures are used to re-sort the data at other delay time and the
final results are shown in Fig. 5(b).

##  VI CONCLUSIONS

In conclusion, we have demonstrated a non-invasive method for time-stamping of
rf compressed electron beams. The method has been used in a pump-probe
experiment to reveal coherent oscillation of A1g phonon mode of Bismuth that
is otherwise smeared out by timing jitter in the raw data. With this method,
the time resolution of rf-compressed UED is greatly improved. This method
requires minimal change (e.g. a bending magnet) to the infrastructure of most
of the UED machines and is applicable to both keV and MeV UED. In our
experiment, the timing jitter after rf compression is corrected with 35 fs
accuracy, limited by the $3\times 10^{-4}$ stability of the rf amplitude in
the gun. For keV UED with greatly improved energy stability, sub-10 fs
accuracy in time-sorting should be readily achievable. It should be noted that
this time-sorting method has large dynamic range and picosecond-level jitter
may be corrected with similar accuracy. Therefore, it may be used to correct
long-term timing drift in rf-compressed UED as well. We expect this method to
have a strong impact in rf-compressed UED where the enhanced resolution by
jitter correction may open up new opportunities in ultrafast science.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grants No. 11925505, 11504232, 12005132 and 11721091), China National
Postdoctoral Program for Innovative Talents (No. BX20200220), and the office
of Science and Technology, Shanghai Municipal Government (No. 16DZ2260200 and
18JC1410700). The data that support the findings of this study are available
from the corresponding author upon reasonable request.

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