# d-sequence edge binomials, and regularity of powers of binomial edge ideals
of trees

Marie Amalore Nambi  Department of Mathematics, IIT Hyderabad, Kandi,
Sangareddy - 502285<EMAIL_ADDRESS>and  Neeraj Kumar  Department
of Mathematics, IIT Hyderabad, Kandi, Sangareddy - 502285
<EMAIL_ADDRESS>
###### Abstract.

We provide the necessary and sufficient conditions for the edge-binomials of
the tree forming a $d$-sequence in terms of the degree sequence notion of a
graph. We study the regularity of powers of the binomial edge ideals of trees
generated by $d$-sequence edge binomials.

###### Key words and phrases:

Binomial edge ideal, Regularity, Tree, $d$-sequence, edge binomial, Degree
sequence.

###### 2020 Mathematics Subject Classification:

Primary 13D02 ; Secondary 05E40

## Introduction

Let $S=k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ be a polynomial ring over a
field $k$. Let $G$ be a finite simple graph on $[n]$ vertices. Given an edge
$\\{i,j\\}$ in $G$, associate a polynomial $f_{ij}=x_{i}y_{j}-x_{j}y_{i}$ in
$S$. We call $f_{ij}$ an _edge binomial_. Denote by $J_{G}\subset S$ the
binomial edge ideal generated by edge binomials.

The binomial edge ideal naturally arises in commutative algebra, algebraic
geometry, and statistics. For instance, Herzog et al. noticed that the
binomial edge ideal appears in the study of conditional independence ideals
suitable to investigate robustness theory in the context of algebraic
statistics (cf. [7]). The notion of binomial edge ideal associated with a
finite simple graph was coined by Herzog et al. in [7] and independently by
Ohtani in [18]. There is a one-to-one correspondence between the binomial edge
ideal of a finite simple graph and set of $2$-minors of $2\times n$ matrix of
indeterminates. Diaconis, Eisenbud, and Sturmfels studied the ideal generated
by all the adjacent $2$-minors of a $2\times n$ generic matrix (cf. [3]). One
can see that the binomial edge ideal of path graphs coincides with the ideal
of adjacent minors of a $2\times n$ matrix of indeterminates. In [3] the
authors also studied corner minors of a $2\times n$ matrix of indeterminates
which coincides with the binomial edge ideal of star graphs. If $G$ is a
complete graph, then $S/J_{G}$ can be visualized as a Segre variety given by
the image of Segre product $\mathbb{P}^{1}\times\mathbb{P}^{n-1}$ of
projective spaces.

In [22], Villarreal proved that edge ideals are of linear type if and only if
the graph is a tree or has a unique cycle of odd length. The binomial edge
ideal of paths (ideal of adjacent minors) is generated by regular sequences
(cf. [5]). The binomial edge ideal of star graphs (ideal of corner minors) is
generated by $d$-sequence (cf. [13]). Since _complete intersection ideal_
$\implies$ _d-sequence ideal_ $\implies$ _ideal of linear type_ (cf. [8]), the
binomial edge ideal of paths and star graphs are of linear type. It is natural
to ask for combinatorial characterization of linear type binomial edge ideals.
Note that there is a strict inclusion among the family of ideals, that is,
_almost complete intersection ideal_ $\implies$ _d-sequence ideal_ $\implies$
_ideal of linear type_ (cf. [8, 13] and [9, page no. 341].).

The notion of _d-sequence_ was introduced by Huneke in [8] (see Definition
1.1). An ideal is said to be generated by $d$-sequence if there is a
generating set that forms a $d$-sequence. In an attempt to find linear type
binomial edge ideals, Jayanthan et al. in [13] characterized graphs whose
binomial edge ideals are almost complete intersection ideals, a subfamily of
linear type ideals. They proposed the following:

######  Conjecture 0.1.

[13, Conjecture 4.17] If the given graph is a tree or a unicyclic graph, then
the binomial edge ideal is of linear type.

In [15], Kumar characterized linear type binomial edge ideal for closed
graphs, namely $J_{G}$ is of linear type if and only if $G$ is a $K_{4}$-free
graph. In this article, we classify all trees (in terms of degree sequence)
whose edge binomials form a $d$-sequence. Our first main result is the
following.

######  Theorem 0.1.

Let $G$ be a tree on vertices $x_{1},\ldots,x_{n}$. Let
$S=k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ be a polynomial ring. Then the
edge binomials of $G$ form a $d$-sequence in $S$ if and only if $G$ has degree
sequences of the following form

  * (i)

$(1,1)$;

  * (ii)

$(m,2,\dots,1)$ or $(m,1,\dots,1)$, where $2\leq m<n$,

  * (iii)

$(m,3,2,\dots,1)$ or $(m,3,1,\dots,1)$ such that the vertex of degree $m$ and
$3$ are adjacent in $G$ with $3\leq m<n$.

Moreover, the binomial edge ideal $J_{G}$ is of linear type.

The above theorem is significant for at least two reasons: Firstly, it
provides a positive answer to a Conjecture 0.1 to some extent; secondly, the
$d$-sequence notion plays a vital role in studying the regularity of powers of
an ideal.

The second part of this article is dedicated to applying $d$-sequence theory
to study the Castelnuovo-Mumford regularity of the binomial edge ideals and
their powers. Several authors have studied the regularity of the binomial edge
ideal of various classes (see, [6, 11, 15, 16, 19, 20, 23] for a partial
list). Several researchers have explored the regularity of the product of
monomial ideals (for instance, see [1] and the reference of this paper). As
part of a technical step, we obtained the regularity of the product of
binomial edge ideals of disjoint union of paths and a complete graph in
Theorem 3.1. The study of regularity and its combinatorial interpretation for
the index of stability is a central research topic in combinatorial
commutative algebra. It is well known that the regularity of powers of any
homogeneous ideal is a linear function, see [2, 14]. It is also known that if
an ideal is generated by a $d$-sequence of $t$ forms of the same degree $r$,
then for all $s\geq t+1$, $reg(I^{s})=(s-t-1)r+reg(I^{t+1}),$ in [2]. Thus
computing the regularity of $I^{s}$, where $I$ is generated by a $d$-sequence
and $s>1$, boils down to computing $\operatorname{reg}(I^{t+1})$. Moreover, if
an ideal $I\subset S$ generated by a $d$-sequence $u_{1},\ldots,u_{n}$, and
set $u_{0}=0\in S$, then one has
$((u_{0},u_{1},\ldots,u_{i-1})+I^{s}):u_{i}=((u_{0},u_{1},\ldots,u_{i-1}):u_{i})+I^{s-1}$,
for $s\geq 1$ and $i=1,\ldots,n$ (see, [21, Observation 2.4]).

A subset $U$ of $V(G)$ is said to be clique if $G[U]$ is a complete graph. A
vertex $v$ of $G$ is said to be a free vertex in $G$ if $v$ is contained in
only one maximal clique. A vertex of $G$ is called an _internal vertex_ if it
is not a free vertex of $G$. Given a tree whose edge binomials form a
$d$-sequence, we express the regularity of $J_{G}^{s}$ in terms of $s$ and the
number of internal vertices of $G$. The second main result is below.

######  Theorem 0.2.

Let $G$ be a tree on vertices $x_{1},\ldots,x_{n}$. Let
$S=k[x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}]$ be a polynomial ring and let
$J_{G}$ be the binomial edge ideal. Let $i(G)$ denote the number of internal
vertices of $G$.

  * (i)

If $G$ has a degree sequence $(m,2,\dots,1)$ or $(m,1,\dots,1)$, then
$\operatorname{reg}{S}/{J_{G}^{s}}=2s+i(G)-1$, for all $s\geq 1$, and $m\geq
2$.

  * (ii)

If $G$ has a degree sequence $(m,3,2,\dots,1)$ or $(m,3,1,\dots,1)$ such that
the vertex of degree $m$ and $3$ are adjacent in $G$, then
$\operatorname{reg}{S}/{J_{G}^{s}}=2s+i(G)-1$, for all $s\geq 1$ and $m\geq
3$.

Note that the stability index is one for each case in the above theorem. In
the case of binomial edge ideals, a few results are known in the literature
about the $\operatorname{reg}J_{G}^{s}$ for $s>1$. To our knowledge, some
results known in this direction are the following. In [12], the authors have
computed the regularity of powers of binomial edge ideals of cycles and star
graphs and obtained the bound for trees and unicycle graphs whose $J_{G}$ is
an almost complete intersection ideal. In [21], the authors obtained the
precise bound for the regularity of $J_{G}^{s}$ for an almost complete
intersection ideal. Ene et al. studied the regularity of $J_{G}^{s}$ for
closed graphs in [6]. We expect that the techniques developed in this article
will stimulate further research interest among combinatorial commutative
algebraists to study the regularity of powers of the ideal.

This paper is organized as follows. The following section recalls definitions,
notations, and known results. Section two characterizes trees, where edge
binomials form a $d$-sequence. In Section 3, we compute the regularity of the
product of binomial edge ideals of a disjoint union of paths and a complete
graph. Section 4 studies the regularity of powers of binomial edge ideal of
trees whose edge binomials form a $d$-sequence.

Acknowledgement. The first author is financially supported by the University
Grant Commission, India. The second author is partially supported by the
Mathematical Research Impact Centric Support (MATRICS) grant from Science and
Engineering Research Board (SERB), India.

##  1\. Preliminaries

### Basic notions from graph theory.

Let $G$ be a simple graph on the vertex set $V(G)=[n]$ and edge set $E(G)$. A
_degree sequence_ of a finite simple graph is a non-increasing sequence of its
vertex degrees. The complete graph on $[n]$ is denoted by $K_{n}$. Let $C_{n}$
(and $P_{n}$) denotes the cycle (path) on $[n]$ vertices respectively. The
length of a path or cycle is its number of edges. A graph with precisely one
cycle as a subgraph is called a unicyclic graph. If a connected graph does not
contain a cycle, it is called a tree. A star graph on $m+1$ vertices, denoted
by $K_{1,m}$, is the graph with vertex set
$V(K_{1,m})=\\{v\\}\sqcup\\{u_{1},\ldots,u_{m}\\}$ and edge set
$E(K_{1,m})=\\{\\{v,u_{i}\\}\mid 1\leq i\leq m\\}$, and we call the vertex $v$
the center of $K_{1,m}$. The degree of a vertex $v\in V(G)$, denoted by
$deg_{G}(v)$, is the number of edges incident to $v$. A vertex $v$ is said to
be a pendant vertex if $\deg_{G}(v)=1$. The distance between two vertices in a
connected graph is the shortest length of a path between two vertices.

######  Definition 1.1.

[10, Definition 1.1] Let $S$ be a commutative ring. Set $a_{0}=0$. A sequence
of elements $a_{1},\ldots,a_{m}$ in $S$ is said to be a $d$-sequence if it
satisfies the two conditions: $a_{1},\ldots,a_{r}$ is a minimal system of
generators of the ideal $I=(a_{1},\ldots,a_{r})$; and $(\langle
a_{0},\ldots,a_{i}\rangle:a_{i+1}a_{j})=(\langle
a_{0},\ldots,a_{i}\rangle:a_{j})$ for all $0\leq i\leq m-1$, and $j\geq i+1$.

######  Remark 1.1.

There are graphs for which edge binomials do not form a $d$-sequence, but some
other minimal generating sets of the binomial edge ideal $J_{G}$ form a
$d$-sequence in $S$. For instance, let $G$ be a unicyclic graph obtained by
attaching a path to each vertex of $C_{3}$. Then $J_{G}$ is generated by
$d$-sequence [13, Theorem 4.4]. However, the edge binomials of $G$ do not form
a $d$-sequence [13, Remark 4.12].

######  Notation 1.1.

Let $G$ be a simple graph on $[n]$. For an edge $e^{\prime}$ in $G$,
$G\setminus e^{\prime}$ is the graph on the vertex set $V(G)$ and edge set
$E(G)\setminus{e}$. An edge $e^{\prime}$ is called a _bridge_ if
$c(G)<c(G\setminus e^{\prime})$, where $c(G)$ is the number of components of
$G$. For a vertex $v$,

$N_{G}(v)=\\{u\in V(G)\mid\\{u,v\\}\in E(G)\\}$

denotes the _neighborhood_ of $v$ in $G$. Let $e=\\{i,j\\}\notin E(G)$ be an
edge in $G\cup\\{e\\}$. Then $G_{e}$ (cf. [17, Definition 3.1]) is the graph
on vertex set $V(G)$ and edge set

$E(G_{e})=E(G)\cup\left\\{\\{k,l\\}:k,l\in N_{G}(i)\textnormal{ or }k,l\in
N_{G}(j)\right\\}.$

We recall some results from the literature relevant to this article. Due to
Mohammadi and Sharifan, the following results describe the colon ideal
operation on the binomial edge ideal.

######  Remark 1.2.

[17, Theorem 3.4] Let $G$ be a simple graph. Let $e=\\{i,j\\}\notin E(G)$ be a
bridge in $G\cup\\{e\\}$. Then $J_{G}:f_{e}=J_{G_{e}}$.

######  Remark 1.3.

Note that for a simple graph $G$, if $e=\\{i,j\\}\notin E(G)$ with
$1\leq\deg_{G\cup e}(i),\deg_{G\cup e}(j)\leq 2$ then $G_{e}=G$, since $\mid
N_{G}(i)\mid\leq 1$ and $\mid N_{G}(j)\mid\leq 1$.

######  Lemma 1.1.

[21, Observation 2.4] Suppose that $u_{1},\ldots,u_{n}$ form a $d$-sequence in
$S$ and $I=\langle u_{1},\ldots,u_{n}\rangle$ is the ideal in $S$. Set
$u_{0}=0$. Then, one has

$((u_{0},u_{1},\ldots,u_{i-1})+I^{s}):u_{i}=((u_{0},u_{1},\ldots,u_{i-1}):u_{i})+I^{s-1},$

for $s\geq 1$ and $i=1,\ldots,n$.

######  Definition 1.2.

Let $S$ be a standard graded polynomial ring over a field $k$. Let $M$ be a
finitely generated graded $S$-module. Let $\mathbf{F}_{\bullet}$ be a minimal
graded $S$-free resolution of $M$:

$\mathbf{F}_{\bullet}:F_{n}\stackrel{{\scriptstyle\phi_{n}}}{{\longrightarrow}}F_{n-1}\longrightarrow\cdots\longrightarrow
F_{1}\stackrel{{\scriptstyle\phi_{1}}}{{\longrightarrow}}F_{0}\longrightarrow
0.$

Here, $F_{i}=\oplus_{j}S(-j)^{\beta_{i,j}}$, where $S(-j)$ denotes the graded
free module of rank $1$ obtained by shifting the degrees in $S$ by $j$, and
$\beta_{i,j}$ denotes the $(i,j)$-th graded Betti number of $M$ over $S$. The
_Castelnuovo-Mumford regularity_ or simply _regularity_ of $M$ over $S$,
denoted by $\operatorname{reg}_{S}M$, is defined as

$\operatorname{reg}_{S}M\coloneqq\max\\{j-i\mid\beta_{i,j}\neq 0\\}.$

For convenience, we shall use $\operatorname{reg}M$ instead of
$\operatorname{reg}_{S}M$. We state the following regularity lemma [4,
Corollary 20.19].

######  Lemma 1.2 (Regularity lemma).

Let $0\rightarrow M\rightarrow N\rightarrow P\rightarrow 0$ be a short exact
sequence of finitely generated graded $S$-modules. Then the following holds.

  1. (a)

$\operatorname{reg}{N}\leq\max\\{\operatorname{reg}M,\operatorname{reg}P\\}$.
The equality holds if $\operatorname{reg}M\neq\operatorname{reg}P+1$.

  2. (b)

$\operatorname{reg}{M}\leq\max\\{\operatorname{reg}N,\operatorname{reg}P+1\\}$.
The equality holds if $\operatorname{reg}N\neq\operatorname{reg}P$.

  3. (c)

$\operatorname{reg}{P}\leq\max\\{\operatorname{reg}M-1,\operatorname{reg}N\\}$.
The equality holds if $\operatorname{reg}M\neq\operatorname{reg}N$.

######  Remark 1.4.

Let $G$ be a new graph obtained by gluing finitely many graphs at free
vertices. Let $G=H_{1}\cup\cdots\cup H_{k}$ be a graph satisfying the
properties

  1. (a)

For $i\neq j,$ if $H_{i}\cap H_{j}\neq\emptyset$, then $H_{i}\cap
H_{j}=\\{v_{ij}\\}$, for some vertex $v_{ij}$ which is free vertex in $H_{i}$
as well as $H_{j}$;

  2. (b)

For distinct $i,j,k,H_{i}\cap H_{j}\cap H_{k}=\emptyset.$

Then $\operatorname{reg}S/J_{G}=\sum_{i=1}^{k}\operatorname{reg}S/J_{H_{i}}$
[11, Corollary 3.2].

######  Remark 1.5.

Let $G$ be a finite simple graph and $J_{G}$ be its binomial edge ideal in
$S$.

  1. (a)

[19, Theorem 2.1] If $G=K_{n}$, then $\operatorname{reg}(S/J_{G})=1$, for any
$n\geq 3$.

  2. (b)

[20, Theorem 4.1] If $G=K_{1,m}$, then $\operatorname{reg}(S/J_{G})=2$, for
any $m\geq 2$.

  3. (c)

[23, Theorem 27] [11, Corollary 3.4] Let $i(G)$ be the number of internal
vertices of $G$. If $G$ has a degree sequence either $(m,2,\ldots,1)$ or
$(m,1,\ldots,1)$, then $\operatorname{reg}S/J_{G}=i(G)+1$.

A graph $G$ is called a _block graph_ if every block of $G$ is a complete
graph. A _flower graph_ (cf. [16, Definition 3.1]) $F_{h,k}(v)$ is a connected
block graph constructed by joining $h$ copies of the cycle graph $C_{3}$ and
$k$ copies of the bipartite graph $K_{1,3}$ with a common vertex $v$, where
$v$ is one of the free vertices of $C_{3}$ and of $K_{1,3}$, and $cdeg(v)\geq
3$. If $G$ has no ﬂower graphs as induced subgraphs, then $G$ is called
flower-free.

######  Remark 1.6.

[16, Corollary 3.2] Let $G$ be a connected block graph that does not have an
isolated vertex. If $G$ is a flower-free graph, then
$\operatorname{reg}S/J_{G}=i(G)+1$, where $i(G)$ is number of internal
vertices of $G$.

##  2\. d-sequence edge binomials

This section characterizes trees for which edge binomials form a $d$-sequence.

The following lemma would describe a characterization for a tree if the
corresponding edge binomials of the tree were to form a $d$-sequence. This
characterization will play an important role in exploring the colon ideal
operation of edge binomials in the form of graph deformation.

######  Lemma 2.1 (Technical Lemma).

Let $G$ be a tree on $[n+1]$. Assume that $a_{1},\ldots,a_{n}$ are
$d$-sequence edge binomials of $G$, where $a_{k}$ corresponds to an edge
$\\{\alpha_{a_{k}},\beta_{a_{k}}\\}\in E(G)$. Let $J_{i}$ denote the ideal
generated by $\langle a_{1},\ldots,a_{i}\rangle$. Let $H_{i}$ denote the graph
associated to $a_{1},\ldots,a_{i}$.

  1. (a)

If $J_{i}:a_{i+1}=J_{i}$ for all $i\geq 1$, then edge binomials
$a_{1},\ldots,a_{n}$ corresponds to a path.

  2. (b)

If there exists a smallest integer $i$ such that $J_{i}:a_{i+1}\neq J_{i}$,
then
$\\{\alpha_{a_{i+1}},\beta_{a_{i+1}}\\}\cap\\{\alpha_{a_{j}},\beta_{a_{j}}\\}\neq\emptyset$,
for all $j>i+1$. In particular, for all $j>i+1$, one has
$\\{\alpha_{a_{i+1}}\\}\cap\\{\alpha_{a_{j}},\beta_{a_{j}}\\}\neq\emptyset$ or
$\\{\beta_{a_{i+1}}\\}\cap\\{\alpha_{a_{j}},\beta_{a_{j}}\\}\neq\emptyset$.
(Intersection property).

###### Proof.

Proof of (a): From Remark 1.3 it follows that $\deg_{J_{i}}(\alpha_{a_{i+1}})$
and $\deg_{J_{i}}(\beta_{a_{i+1}})$ are less than or equal to $1$, for all
$i$. Thus, all vertices of $G$ have degrees less than or equal to $2$. Since
$G$ is a tree, it follows that $G$ is a path.

Proof of (b): If $i=n-1$, then proof follows immediately. Assume that $i<n-1$.
Since $J_{i}:a_{i+1}\neq J_{i}$, implies that there exists $f_{kl}\in
J_{i}:a_{i+1}$, where $\\{k,l\\}\in N_{H_{i}}(\alpha_{a_{i+1}})$ or
$\\{k,l\\}\in N_{H_{i}}(\beta_{a_{i+1}})$, by Remark 1.2. If $\\{k,l\\}\in
N_{H_{i}}(\alpha_{a_{i+1}})$ then clearly $\\{k,l\\}\notin
N_{H_{i}}(\beta_{a_{i+1}})$, otherwise $G$ has a cycle. So one can further
assume that $\\{k,l\\}\in N_{H_{i}}(\alpha_{a_{i+1}})$. Suppose there exists
an edge binomial $a_{j}$ such that
$\\{\alpha_{a_{i+1}}\\}\cap\\{\alpha_{a_{j}},\beta_{a_{j}}\\}=\emptyset$, and
$\\{\beta_{a_{i+1}}\\}\cap\\{\alpha_{a_{j}},\beta_{a_{j}}\\}=\emptyset$. We
have $J_{i}:a_{i+1}a_{j}=J_{i}:a_{j}$ by hypothesis. This implies that
$f_{lk}\in J_{i}:a_{j}$. Since $f_{kl}\notin J_{i}$ implies that $\\{k,l\\}\in
N_{H_{i}}(\alpha_{a_{j}})$ or $\\{k,l\\}\in N_{H_{i}}(\beta_{a_{j}})$. In both
cases, $G$ has a cycle; this contradicts the fact that $G$ is a tree. For
instance if $\\{k,l\\}\in N_{H_{i}}(\alpha_{a_{j}})$ the edges
$\\{\alpha_{a_{i+1}},k\\},\\{\alpha_{a_{i+1}},l\\},\\{\alpha_{a_{j}},k\\},\\{\alpha_{a_{j}},l\\}$
forms a cycle. Thus
$\\{\alpha_{a_{i+1}}\\}\cap\\{\alpha_{a_{j}},\beta_{a_{j}}\\}\neq\emptyset$ or
$\\{\beta_{a_{i+1}}\\}\cap\\{\alpha_{a_{j}},\beta_{a_{j}}\\}\neq\emptyset$,
for all $j>i+1$. Hence the proof is complete. ∎

######  Example 1.

Let $G$ be a tree on $[7]$ with edge set

$E(G)=\\{\\{1,2\\},\\{1,3\\}\\{1,4\\},\\{i,j\\},\\{i_{1},j_{1}\\},\\{i_{2},j_{2}\\}\\}.$

Let
$a_{1}=f_{12},a_{2}=f_{13},a_{3}=f_{14},a_{4}=f_{ij},a_{5}=f_{i_{1}j_{1}},a_{6}=f_{i_{2}j_{2}}$
be a sequence of edge binomials of $G$. One can see that $(J_{2}:a_{3})\neq
J_{2}$, since $f_{23}\in(J_{2}:a_{3})$. Suppose the sequence
$a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}$ form a $d$-sequence, then the sequence
must satisfy the following condition $J_{2}:a_{3}a_{4}=J_{2}:a_{4}$. Since
$a_{4}$ is a bridge, we can conclude that either $\\{2,3\\}\in N_{H_{2}}(i)$
or $\\{2,3\\}\in N_{H_{2}}(j)$. Assume that $\\{2,3\\}\in N_{H_{2}}(i)$.

Case (i): If $i=1$ then it is obvious that
$\\{1\\}\cap\\{i,j\\}\neq\emptyset$.

Case (ii): If $i\neq 1$ then $\\{1,2\\},\\{2,i\\},\\{i,3\\},\\{3,1\\}$ form a
cycle in $G$. That is not possible since $G$ is a tree. Therefore, we may
conclude that $\\{1\\}\cap\\{i,j\\}\neq\emptyset$.

######  Notation 2.1.

Let $\mathcal{T}_{m}$ denote the class of graphs having degree sequence in
either of the form $(m,2,\ldots,1)$ or $(m,1,\ldots,1)$, where $m\geq 2$.

Equivalently, one can describe $G\in\mathcal{T}_{m}$ in terms of a vertex set
and edge set as below:

$V(G)=\\{k_{0},p_{1_{1}},\ldots,p_{1_{s(1)+1}},p_{2_{1}},\ldots,p_{2_{s(2)+1}},\ldots,p_{m_{1}},\ldots,p_{m_{s(m)+1}}\\}$

with $s(i)\geq 0$ for all $1\leq i\leq m$, and edge set

$E(G)=\\{\\{k_{0},p_{i_{1}}\mid
i=1,\ldots,m\\}\cup\bigcup_{i=1}^{m}\\{p_{i_{j}},p_{i_{j+1}}\mid
j=1,\ldots,s(i)\\}\\}.$

Let $\mathcal{P}$ be an induced subgraph of $G$ on $V(G)\setminus k_{0}$. We
call $\mathcal{P}$ as paths of $G$ and the vertex $k_{0}$ as the center of
$G$. Let $\ell$ denote the number of vertices of degree $2$ in $G$. Notice
that $i(G)=\sum_{i=1}^{m}s_{(i)}+1$, for all $m\geq 2$.

The degree sequence presentation helps us to visualize the graph and the
second description is helpful for proving results from the notation point of
view. For illustration purpose, a graph $G_{1}\in\mathcal{T}_{5}$ having
degree sequence $(5,2,2,2,1,1,1,1,1)$ is shown in Figure 1.

$k_{0}$$p_{1_{1}}$$p_{1_{2}}$$p_{2_{1}}$$p_{2_{2}}$$p_{3}{{}_{1}}$$p_{4_{1}}$$p_{4_{2}}$$p_{5_{1}}$
Figure 1. $G_{1}$

######  Theorem 2.1.

Let $G$ be a $\mathcal{T}_{m}$ graph on $[n]$. The edge binomials of $G$ form
a $d$-sequence.

###### Proof.

Set $d_{0}=0\in S$. We claim $G$ is generated by the $d$-sequence
$d_{1},\ldots,d_{n-1}$, where the first $n-m-1$ elements are edge binomials of
$\mathcal{P}$ (edge binomials of paths of $G$ in independent of order) and

$d_{i}=x_{k_{0}}y_{{p_{i-(n-m-1)}}_{1}}-x_{{p_{i-(n-m-1)}}_{1}}y_{k_{0}},\text{
for }n-m-1<i\leq n-1.$

Then for all $0\leq i\leq n-m+1$, and for all $j\geq i+1$,

$\begin{split}(d_{0},d_{1},\ldots,d_{i}):d_{i+1}d_{j}&=((d_{0},d_{1},\ldots,d_{i}):d_{i+1}):d_{j}\\\
&=(d_{0},d_{1},\ldots,d_{i}),\end{split}$

also

$(d_{0},d_{1},\ldots,d_{i}):d_{j}=(d_{0},d_{1},\ldots,d_{i}),$

where the equality follows from Remark 1.3, since the graphs associated to the
binomial edge ideal $(d_{0},d_{1},\ldots,d_{i},d_{i+1})$ and
$(d_{0},d_{1},\ldots,d_{i},d_{j})$ are disjoint union of paths. Also for all
$n-m+1<i\leq\ n-2$, and for all $j\geq i+1$,

(2.1)
$\begin{split}(d_{0},d_{1},\ldots,d_{i}):d_{i+1}d_{j}&=((d_{0},d_{1},\ldots,d_{i}):d_{i+1}):d_{j}\\\
&=(d_{0},d_{1},\ldots,d_{i})+\langle f_{kl}\mid\\{k,l\\}\in
N(k_{0})\rangle,\end{split}$

and moreover,

(2.2) $(d_{0},d_{1},\ldots,d_{i}):d_{j}=(d_{0},d_{1},\ldots,d_{i})+\langle
f_{kl}\mid\\{k,l\\}\in N(k_{0})\rangle,$

where the equality in (2.1) and (2.2) follows from Remark 1.2 and Lemma 2.1.
Thus, we get
$(d_{0},d_{1},\ldots,d_{i}):d_{i+1}d_{j}=((d_{0},d_{1},\ldots,d_{i}):d_{j}$,
for all $0<i\leq\ n-2$, and for all $j\geq i+1$. Therefore, the edge binomials
of $G$ graph form a $d$-sequence. ∎

######  Corollary 2.1.

Let $G$ be a $\mathcal{T}_{m}$ graph on $[n]$. Then $J_{G}$ is of linear type.

###### Proof.

The proof follows from [8, Theorem 3.1]. ∎

######  Notation 2.2.

Let $\mathcal{H}_{m}$ be a class of trees having a degree sequence in either
of the form $(m,3,2,\ldots,1)$ or $(m,3,1,\ldots,1)$ such that the vertex of
degree $m$ and $3$ are adjacent, where $m\geq 3$.

Equivalently, one can describe $G\in\mathcal{H}_{m}$ in terms of a vertex set
and edge set as below:

$V(G)=\\{k_{0},k_{1},p_{1_{1}},\ldots,p_{1_{s(1)+1}},p_{2_{1}},\ldots,p_{2_{s(2)+1}},\ldots,p_{{m+1}_{1}},\ldots,p_{{m+1}_{s(m+1)+1}}\\}$

with $s(i)\geq 0$ for all $1\leq i\leq m+1$, and edge set

$\begin{split}E(G)=&\\{\\{k_{0},p_{i_{1}}\mid
i=1,\ldots,m-1\\}\cup\\{k_{0},k_{1}\\}\cup\\{k_{1},p_{i_{1}}\mid i=m,m+1\\}\\\
&\bigcup_{i=1}^{m+1}\\{p_{i_{j}},p_{i_{j+1}}\mid
j=1,\ldots,s(i)\\}\\}.\end{split}$

For illustration purpose, a graph $G_{2}\in\mathcal{H}_{4}$ having degree
sequence $(4,3,1,1,1,1,1)$ is shown in Figure 2. Notice that
$i(G)=\sum_{i=1}^{m+1}s_{(i)}+2$, for all $m\geq 3$, where
$G\in\mathcal{H}_{m}$.

$k_{0}$$p_{1_{1}}$$p_{2_{1}}$$p_{3_{1}}$$k_{1}$$p_{5_{1}}$$p_{4_{1}}$ Figure
2. $G_{2}$

######  Theorem 2.2.

Let $G$ be a $\mathcal{H}_{m}$ graph on $[n]$. The edge binomials of $G$ form
a $d$-sequence. In particular, $J_{G}$ is of linear type.

###### Proof.

Consider the following edge binomial sequence $d_{1},\ldots,d_{n-1}$, where
the first $n-m-1$ elements are edge binomials of $\mathcal{P}$, where
$\mathcal{P}$ denotes an induced subgraph of $G$ on $V(G)\setminus k_{0}$
(independent of order),

$d_{i}=x_{k_{0}}y_{p_{{i-(n-m-1)}_{1}}}-x_{p_{{i-(n-m-1)}_{1}}}y_{k_{0}},$

for $n-m-1<i\leq n-2$, and

$d_{n-1}=x_{k_{0}}y_{k_{1}}-x_{k_{1}}y_{k_{0}}.$

The rest of the proof is similar to the proof of Theorem 2.1. ∎

We recall a couple of definitions from [21]. A graph $G$ is of $T$-type graph
if $G$ is obtained by adding an edge between an internal vertex of a path and
a pendant vertex of another path. A graph $G$ is said to be a $H$-type graph
if $G$ is obtained by adding an edge between two internal vertices of two
distinct paths.

######  Remark 2.1.

The authors in [13, Theorem 4.3.] have characterized trees whose binomial edge
ideals are almost complete intersections. Also, the authors proved that almost
complete intersection binomial edge ideals are generated by a $d$-sequence.
Shen and Zhu [21] classified almost complete intersection binomial edge ideals
into two types of graphs called $T$-type and $H$-type. One can see that the
$T$-type graph is an induced subgraph of $\mathcal{T}_{m}$ graph, and the
$H$-type graphs are an induced subgraph of $\mathcal{H}_{m}$ graph.

######  Proposition 2.1.

If edge binomials of tree form a $d$-sequence, then the tree is one of the
following graph $P_{2}$, or $\mathcal{T}_{m}$ graph, or $\mathcal{H}_{m}$
graph.

###### Proof.

Let $G$ be tree on $[n]$ such that edge binomials $d_{1},\ldots,d_{n-1}$ of
$G$ form a $d$-sequence. Suppose there exists the smallest integer $i$ such
that $J_{i}:d_{i+1}\neq J_{i}$, then for all $j\leq i$, edge binomials
$d_{1},\ldots,d_{j}$ corresponds to the edges of disjoint union of paths and
the remaining edge binomials $\\{d_{l}\mid i<l\leq n-1\\}$ corresponds to the
edges of $K_{1,m}$ for some $m$ by Lemma 2.1. Note that the graph associated
with the binomial edge ideal generated by $d_{1},\ldots,d_{i+1}$, have
$K_{1,3}$ as a subgraph. Denote the center of $K_{1,3}$ by a vertex $k_{0}$.
From Lemma 2.1, it follows that all the edges corresponding to $d_{l}$ for all
$i+1\leq l\leq n-1$, intersects with $k_{0}$.

Thus $G$ can be obtained by identifying one free vertex of each path with some
free vertices of $K_{1,m}$ since $G$ is a tree. Observe that one can identify
an internal vertex of a path with a pendant vertex of $K_{1,m}$ in that case,
we consider it as two free vertices of two distinct paths identified with a
pendant vertex of $K_{1,m}$.

Case 1: If one free vertex of each path is identified with distinct pendant
vertices of $K_{1,m}$, then $G$ is isomorphic to a graph in $\mathcal{T}_{m}$.

Case 2: Let $k_{1}$ be a pendant vertex of $K_{1,m}$. Suppose the vertex
$k_{1}$ is identified with two free vertices of two distinct paths. One free
vertex from each path is identified from the remaining paths with distinct
pendant vertices of $K_{1,m}\setminus k_{1}$. In that case, $G$ is isomorphic
to a graph in $\mathcal{H}_{m}$.

Case 3: If three (or more) pendant vertices of three (or more) distinct paths
are identified with a pendant vertex of $K_{1,m}$ then it contradicts the
choice of the smallest integer $i$.

Case 4: Let $k_{1}$ and $k_{2}$ be two pendant vertices of $K_{1,m}$. Let
$d_{p}=f_{k_{0}k_{1}}$ and $d_{q}=f_{k_{0}k_{2}}$ be edge binomials, where
$p,q>i$. Suppose the vertex $k_{1}$ is identified with two free vertices of
two distinct paths and $k_{2}$ is identified with two free vertices of another
two distinct paths, then $J_{p-1}:d_{p}d_{q}\neq J_{p-1}:d_{q}$, where $p<q$,
by Remark 1.2. Therefore, in this case, any sequence of edge binomials of $G$
does not satisfy the $d$-sequence condition.

If no such $i$ exists, then from Lemma 2.1(a) it follows that $G$ is a path. ∎

As an immediate consequence, we have the following:

######  Corollary 2.2.

Let $G$ be a tree. If $G$ has at least two vertices with degrees greater than
or equal to $4$ or $G$ has at least three vertices with degrees greater than
or equal to $3$, then any sequence of edge binomials of $G$ does not form a
$d$-sequence.

######  Corollary 2.3.

Let $G$ be a tree. If there exists $u,v\in V(G)$ such that
$\deg({u}),\deg({v})\geq 3$ and $d(u,v)\geq 2$. Then any sequence of edge
binomials of $G$ does not form a $d$-sequence.

### Conclusion

The proof of Theorem 0.1 follows from Theorems 2.1, 2.2 and Proposition 2.1.

##  3\. Regularity of the product of a disjoint union of paths and a complete
graph

In this section, we state the regularity of the binomial edge ideal of
$d$-sequence graphs. The results of this section will be used in Section $4$.

######  Lemma 3.1.

Let $G$ be a tree. If $G\in\\{\mathcal{T}_{m},\mathcal{H}_{m}\\}$ graph, then
$\operatorname{reg}{S}/{J_{G}}=i(G)+1$.

###### Proof.

Observe that any $G\in\\{\mathcal{T}_{m},\mathcal{H}_{m}\\}$ is a flower-free
block graphs. Thus the result follows from Remark 1.6. ∎

######  Notation 3.1.

Let $n\geq 3$, and $m\geq 1$. $\mathcal{C}_{n,m}$ be a graph on $[n+m]$ such
that

$V(\mathcal{C}_{n,m})=\\{v_{0},v_{1},\ldots,v_{n-1},k_{1},\ldots,k_{m}\\}$

and edge set

$E(\mathcal{C}_{n,m})=\\{\\{v_{i},v_{j}\mid 0\leq i<j\leq
n-1\\},\\{v_{0},k_{i}\mid 1\leq i\leq m\\}\\}.$

The $\mathcal{C}_{3,2}$ graph is illustrated in Figure 3.

$v_{0}$$k_{1}$$k_{2}$$v_{1}$$v_{2}$ Figure 3. $\mathcal{C}_{3,2}$

######  Lemma 3.2.

Let $G$ be a $\mathcal{C}_{n,m}$ graph. Then
$\operatorname{reg}{S}/{J_{G}}=2$, for any $m\geq 1$ and $n\geq 3$.

###### Proof.

The proof follows from Remark 1.6. ∎

Next, we obtain the regularity of the product of binomial edge ideals of
disjoint union of paths and a complete graph.

######  Theorem 3.1.

Let $H=\\{P_{1}^{\prime},\ldots P_{t}^{\prime}\\}$ be a disjoint union of
paths and $K_{m}$ be a complete graph on $[m]$ such that

  1. (1)

for any $i$, if $K_{m}\cap P_{i}^{\prime}\neq\emptyset$, then $V(K_{m})\cap
V(P_{i}^{\prime})=v_{i}$, for some $v_{i}$ which is free vertex in
$P_{i}^{\prime}$

  2. (2)

$V(K_{m})\cap V(P_{i}^{\prime})\cap V(P_{j}^{\prime})=\emptyset$, for all
distinct $i$ and $j$.

Let $n$ be the number of edges in $H$. Then, for any $n\geq 1$ and for any
$m\geq 2$, we have

$\operatorname{reg}\frac{S}{J_{H}J_{K_{m}}}=2+n.$

###### Proof.

Let $J_{H}=\langle a_{1},\ldots,a_{n}\rangle$ be the binomial edge ideal of
the disjoint union of paths and $J_{K_{m}}$ be the binomial edge ideal of a
complete graph on $[m]$. Consider the following set of short exact sequences:

(3.1)
$0\longrightarrow\frac{S}{(J_{H}J_{K_{m}}:a_{1})}(-2)\longrightarrow\frac{S}{J_{H}J_{K_{m}}}\longrightarrow\frac{S}{(J_{H}J_{K_{m}},a_{1})}\longrightarrow
0;$ $\vdots$ (3.2)
$0\longrightarrow\frac{S}{((J_{H}J_{K_{m}},a_{1},\ldots,a_{n-1}):a_{n})}(-2)\longrightarrow\frac{S}{(J_{H}J_{K_{m}},a_{1},\ldots,a_{n-1})}\longrightarrow\frac{S}{J_{H}}\longrightarrow
0.$

Note that,
$((J_{H}J_{K_{m}},a_{1},\ldots,a_{r-1}):a_{r})=(J_{K_{m}},a_{1},\ldots,a_{r-1})$,
for any $1\leq r\leq n$. Clearly the inclusion
$((J_{H}J_{K_{m}},a_{1},\ldots,a_{r-1}):a_{r})\supseteq(J_{K_{m}},a_{1},\ldots,a_{r-1})$,
holds. Since for every $j\in J_{K_{m}}$, we have $ja_{r}\in J_{H}J_{K_{m}}$.
For other side inclusion, let
$x\in((J_{H}J_{K_{m}},a_{1},\ldots,a_{r-1}):a_{r})$. Then
$xa_{r}\in(J_{H}J_{K_{m}},a_{1},\ldots,a_{r-1})$ implies that

$xa_{r}=\sum_{k_{i}\in K_{m}}r_{i}k_{i}a_{r}+\sum_{k_{i}\in K_{m},r<j\leq
n}s_{ij}k_{i}a_{j}+\sum_{1\leq i\leq r-1}t_{i}a_{i},$

for some $r_{i},s_{ij},t_{i}\in S$. One can conclude that $s_{ij}\text{ and
}t_{i}\in\langle a_{r}\rangle$, since $J_{G}=J_{G^{\prime}}$ if and only if
$E(G)=E(G^{\prime})$. Thus, it follows that
$x\in(J_{K_{m}},a_{1},\ldots,a_{r-1})$.

From the hypothesis, it follows that the binomial edge ideal
$(J_{K_{m}},a_{1},\ldots,a_{r-1})$ is associated with the graph obtained by
gluing $K_{m}$ and paths at free vertices, and disjoint union of paths.
Therefore, using Remark 1.4, we get

$\operatorname{reg}\frac{S}{(J_{K_{m}},a_{1},\ldots,a_{r-1})}=1+r-1,$

for any $1\leq r\leq n$. Since from Remark 1.5(a) it follows that
$\operatorname{reg}{S}/{J_{K_{m}}}=1$. Also it is known that
$\operatorname{reg}{S}/{J_{H}}=n$. From Lemma 1.2(a), it follows that the
regularity of the middle module of short exact sequence (3.2) is

$\operatorname{reg}\frac{S}{(J_{H}J_{K_{m}},a_{1},\ldots,a_{n-1})}=2+n.$

Applying $\operatorname{reg}\frac{S}{(J_{H}J_{K_{m}},a_{1},\ldots,a_{n-1})}$
to the preceding short exact sequence of (3.2), yields

$\operatorname{reg}\frac{S}{(J_{H}J_{K_{m}},a_{1},\ldots,a_{n-2})}=2+n.$

Similarly, one has

$\operatorname{reg}\frac{S}{(J_{H}J_{K_{m}},a_{1},\ldots,a_{r-1})}=2+n,\text{
for all }1\leq r\leq n-2.$

In particular,

$\operatorname{reg}\frac{S}{(J_{H}J_{K_{m}})}=2+n.$

∎

##  4\. Regularity of powers of d-sequence binomial edge ideal

In this section, we obtain precise expressions for the regularity of powers of
the binomial edge ideal of $d$-sequence graphs.

The authors in [11] computed the regularity of binomial edge ideal of
$\mathcal{T}_{m}$ graph. We now calculate the regularity of powers of such
binomial edge ideal.

######  Theorem 4.1.

Let $G$ be a $\mathcal{T}_{m}$ graph on $[n+1]$. Let $d_{1},\ldots,d_{n}$ be a
sequence of edge binomials of $G$ such that $d_{1},\ldots,d_{n}$ forms a
$d$-sequence. Set $d_{0}=0\in S$. Then, for all $s\geq 1$, and $m\geq 2$, and
$i=0,1,\ldots,n-1$, we have

$\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{i})+J_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}.$

In particular, $\operatorname{reg}{S}/{J_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}$,
for all $s\geq 1$.

The edge binomials of $\mathcal{T}_{m}$ graph form a $d$-sequence, and we take
$d_{1},\ldots,d_{n}$ to be in the same order as in Theorem 2.1. To complete
the proof of Theorem 4.1, we need to prove a couple of lemmata.

######  Lemma 4.1.

Under the assumption in Theorem 4.1, and for any $i=0,1,\ldots,n-1$, we have

$\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{i})+J_{G}^{2}}=4+\sum_{j=1}^{m}{s_{(j)}}.$

###### Proof.

Consider the following short exact sequence

(4.1)
$0\longrightarrow\frac{S}{(d_{1},\ldots,d_{n-1}):d_{n}}(-2)\longrightarrow\frac{S}{(d_{1},\ldots,d_{n-1})}\longrightarrow\frac{S}{(d_{1},\ldots,d_{n})}\longrightarrow
0.$

From Remark 1.2, it follows that ideal $((d_{1},\ldots,d_{n-1}):d_{n})$ is
binomial edge ideal of graph obtained by gluing ${K_{m}}$ and paths at free
vertices. Thus from Remark 1.4 it follows that
$\operatorname{reg}{S}/{((d_{1},\ldots,d_{n-1}):d_{n})}=1+\sum_{j=1}^{m}s_{(j)}$.
From Remark 1.5(c) it follows that
$\operatorname{reg}{S}/{J_{G}}=2+\sum_{j=1}^{m}s_{(j)}$. Applying Lemma 1.2(a)
to the exact sequence (4.1), yields that

$\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{n-1})}\leq
2+\sum_{j=1}^{m}s_{(j)}.$

Now, the proof of the lemma is by descending induction on $i$. For $i=n-1$,
Consider the short exact sequence

(4.2)
$0\longrightarrow\frac{S}{(d_{1},\ldots,d_{n-1}):d_{n}^{2}}(-4)\longrightarrow\frac{S}{(d_{1},\ldots,d_{n-1})}\longrightarrow\frac{S}{(d_{1},\ldots,d_{n-1},d_{n}^{2})}\longrightarrow
0.$

Since sequence of edge binomials $d_{1},\ldots,d_{n}$ form a $d$-sequence, one
has $(d_{1},\ldots,d_{n-1}):d_{n}^{2}=(d_{1},\ldots,d_{n-1}):d_{n}$. As
$\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{n-1}):d_{n}^{2}}(-4)=5+\sum_{j=1}^{m}s_{(j)}>\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{n-1})}$,
it follow from Lemma 1.2(c) that

$\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{n-1})+J_{G}^{2}}=\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{n-1})+d_{n}^{2}}=4+\sum_{j=1}^{m}s_{(j)}.$

Assume that the assertion holds for $j$. Consider the short exact sequence

(4.3)
$\begin{split}0\longrightarrow\frac{S}{((d_{1},\ldots,d_{j-1})+J_{G}^{2}):d_{j}}(-2)&\longrightarrow\frac{S}{(d_{1},\ldots,d_{j-1})+J_{G}^{2}}\\\
&\longrightarrow\frac{S}{(d_{1},\ldots,d_{j})+J_{G}^{2}}\longrightarrow
0.\end{split}$

From Lemma 1.1, it follows that
$((d_{1},\ldots,d_{j-1})+J_{G}^{2}):d_{j}=((d_{1},\ldots,d_{j-1}):d_{j})+J_{G}$.
From Remark 1.2, $((d_{1},\ldots,d_{j-1}):d_{j})+J_{G}=J_{G}+\langle
f_{lk}\mid\\{l,k\\}\in N_{H_{i}}(\alpha_{d_{i+1}})$ or $\\{l,k\\}\in
N_{H_{i}}(\beta_{d_{i+1}})\rangle$, is the binomial edge ideal of graph
obtained by gluing $\mathcal{C}_{n,m-n+1}$ and paths at free vertices, where
$n<m$. Thus, from Lemma 3.2, and Remark 1.4, it follows that
$\operatorname{reg}{S}/{(((d_{1},\ldots,d_{j-1}):d_{j})+J_{G})}=2+\sum_{j=1}^{m}s_{(j)}$.
By the induction hypothesis conclude that
$\operatorname{reg}{S}/{((d_{1},\ldots,d_{j})+J_{G}^{2})}=4+\sum_{j=1}^{m}s_{(j)}$.
Then by applying Lemma 1.2(a) to (4.3), we get

$\frac{S}{(d_{1},\ldots,d_{j-1})+J_{G}^{2}}=4+\sum_{j=1}^{m}s_{(j)},$

as desired. ∎

######  Lemma 4.2.

Under the assumption in Theorem 4.1, and for all $s>1$, we have

$\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{n-1})+d_{n}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}.$

###### Proof.

Consider the following short exact sequence of graded modules:

(4.4)
$0\longrightarrow\frac{S}{(d_{1},\ldots,d_{n-1}):d_{n}^{s}}(-2s)\longrightarrow\frac{S}{(d_{1},\ldots,d_{n-1})}\longrightarrow\frac{S}{(d_{1},\ldots,d_{n-1})+d_{n}^{s}}\longrightarrow
0.$

Since $d_{1},\ldots,d_{n}$ form a $d$-sequence, one has
$(d_{1},\ldots,d_{n-1}):d_{n}^{s}=(d_{1},\ldots,d_{n-1}):d_{n}$. From Lemma
4.1, we know the regularity of modules ${S}/{((d_{1},\ldots,d_{n-1}):d_{n})}$
and ${S}/{(d_{1},\ldots,d_{n-1})}$. Then from Lemma 1.2(c) it follows that
$\operatorname{reg}{S}/{((d_{1},\ldots,d_{n-1})+d_{n}^{s})}=2s+\sum_{j=1}^{m}s_{(j)}$.
∎

###### Proof of Theorem 4.1.

The proof is by induction on $s$. For $s=1$, the statement follows from Remark
1.5(c). We may assume that the assertion holds for $s-1$. The proof for the
statement $s$ is by descending induction on $i$. For $i=n-1$, the statement
holds by Lemma 4.2. Next, assume that the assertion holds for $i+1$. Consider
the short exact sequence

(4.5)
$\begin{split}0\longrightarrow\frac{S}{(d_{1},\ldots,d_{i})+J_{G}^{s}:d_{i+1}}(-2)&\longrightarrow\frac{S}{(d_{1},\ldots,d_{i})+J_{G}^{s}}\\\
&\longrightarrow\frac{S}{(d_{1},\ldots,d_{i+1})+J_{G}^{s}}\longrightarrow
0.\end{split}$

By the induction hypothesis on $i$ that
$\operatorname{reg}{S}/{(d_{1},\ldots,d_{i+1})+J_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}$.
By Lemma 1.1 one has
$((d_{1},\ldots,d_{i})+J_{G}^{s}):d_{i+1}=((d_{1},\ldots,d_{i}):d_{i+1})+J_{G}^{s-1}$.
By Remark 1.2, it follows that
$((d_{1},\ldots,d_{i}):d_{i+1})+J_{G}^{s-1}=(d_{1},\ldots,d_{i})+J_{G}^{s-1}+\langle
f_{lk}\mid\\{l,k\\}\in N_{H_{i}}(\alpha_{d_{i+1}})$ or $\\{l,k\\}\in
N_{H_{i}}(\beta_{d_{i+1}})\rangle$. Note that $\langle f_{lk}\mid\\{l,k\\}\in
N_{H_{i}}(\alpha_{d_{i+1}})$ or $\\{l,k\\}\in
N_{H_{i}}(\beta_{d_{i+1}})\rangle=J_{K_{n}}$, for some $n<m$, by Lemma 2.1.
Set $I=(d_{1},\ldots,d_{i})+J_{G}^{s-1}$ and $J=J_{K_{n}}$. Consider the two
cases:

Case 1: If $((d_{1},\ldots,d_{i}):d_{i+1})+J_{G}^{s-1}=I$, then from the
induction hypothesis on $s$, it follows that
$\operatorname{reg}{S}/{I}=2(s-1)+\sum_{j=1}^{m}{s_{(j)}}$. Now, using Lemma
1.2(a) to the short exact sequence (4.5) yields

$\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{i})+J_{G}^{s}}=2s+\sum_{j=1}^{m}{s_{(j)}}.$

Case 2: If $((d_{1},\ldots,d_{i}):d_{i+1})+J_{G}^{s-1}=I+J$, then from the
following short exact sequence

(4.6) $0\longrightarrow\frac{S}{I\cap
J}\longrightarrow\frac{S}{I}\oplus\frac{S}{J}\longrightarrow\frac{S}{I+J}\longrightarrow
0,$

one has,

$\operatorname{reg}{\frac{S}{I+J}}=\max\\{\operatorname{reg}{\frac{S}{I}},\operatorname{reg}{\frac{S}{J}},\operatorname{reg}{\frac{S}{I\cap
J}}-1\\},$

if $\operatorname{reg}{{S}/{(I\cap
J)}}\neq\max\\{\operatorname{reg}{{S}/{I}},\operatorname{reg}{{S}/{J}}\\}$.
From the induction hypothesis on $s$, it follows that
$\operatorname{reg}{S}/{I}=2(s-1)+\sum_{j=1}^{m}{s_{(j)}}$. Since $J$ is a
complete graph, from Remark 1.5(a) it follows that
$\operatorname{reg}{S}/{J}=1$.

######  Claim 4.1.

$I\cap J=J*\langle x_{k_{0}},y_{k_{0}},J_{\mathcal{P}}\rangle$, where
$\mathcal{P}$ denotes the paths of $G$ and $k_{0}$ denote the center of $G$.

For $I\cap J\supseteq J*\langle x_{k_{0}},y_{k_{0}},J_{\mathcal{P}}\rangle$
inclusion, it is enough to show that $\\{f_{lk}x_{k_{0}},f_{lk}y_{k_{0}}\\}\in
I$ for all $f_{lk}\in J$, since $J_{\mathcal{P}}\subset I$. For any $f_{lk}\in
J$ we have $f_{lk_{0}}=d_{i_{1}}$ and $f_{kk_{0}}=d_{i_{2}}$, for some
$i_{1},i_{2}<i$. This implies that $\\{f_{lk_{0}},f_{kk_{0}}\\}\in I$. Then
$x_{l}f_{kk_{0}}-x_{k}f_{lk_{0}}=x_{k_{0}}f_{lk}$ and
$y_{l}f_{kk_{0}}-y_{k}f_{lk_{0}}=y_{k_{0}}f_{lk}$, as desired. For other side
inclusion, for $i<n-1$, let $H_{1}$ be a graph associated to binomials
$d_{1},\ldots,d_{i}$, $H_{2}$ be a graph associated to binomials
$d_{i+1},\ldots,d_{n}$, and $H_{3}$ be a graph such that $E(H_{3})=\\{k,l\\}$
where $\\{k,l\\}\in N_{H_{1}}(\alpha_{d_{i+1}})$ or $\\{k,l\\}\in
N_{H_{1}}(\beta_{d_{i+1}})$. Notice that $E(H_{r_{1}})\cap
E(H_{r_{2}})=\emptyset$ where $r_{1},r_{2}\in\\{1,2,3\\}$,
$I=J_{H_{1}}+J_{H_{2}}^{s-1}$ and $J=J_{H_{3}}$.

Assume that $x\notin J*\langle x_{k_{0}},y_{k_{0}},J_{\mathcal{P}}\rangle$ and
$x\in J$. Thus, $x=\sum_{g_{i}\in J_{H_{3}}}r_{i}g_{i}$, where $r_{i}\in S$.
Now, suppose $x\in I$, then $r_{i}\in I$ for all $i$, since $(E(H_{1})\cup
E(H_{2}))\cap E(H_{3})=\emptyset$. Therefore, $x\in IJ$, but $IJ\subset
J*\langle x_{k_{0}},y_{k_{0}},J_{\mathcal{P}}\rangle$, which is a
contradiction. Thus $x\notin I$. In particular, $x\notin I\cap J$. Now, assume
that $x\notin J*\langle x_{k_{0}},y_{k_{0}},J_{\mathcal{P}}\rangle$ and $x\in
I$. Thus, $x=\sum_{g_{i}\in
J_{H_{1}}}r_{i}g_{i}+\sum_{h_{k}\in{J_{H_{2}}}^{s-1}}t_{k}h_{k},\textnormal{where}\
r_{i},t_{k}\in S$. Suppose $x\in J$, then $r_{i},t_{k}\in J$ for all $i$ and
$k$, since $(E(H_{1})\cup E(H_{2}))\cap E(H_{3})=\emptyset$. Therefore, $x\in
IJ$, but $IJ\subset J*\langle x_{k_{0}},y_{k_{0}},J_{\mathcal{P}}\rangle$,
which is a contradiction. Thus $x\notin J$.

From Theorem 3.1 it follows that $\operatorname{reg}S/(I\cap
J)=2+\sum_{j=1}^{m}{s_{(j)}}$. Then applying regularity lemma to the short
exact sequence (4.6) yields that for all $s>2$,

$\operatorname{reg}{{S}/{(I+J)}}=2(s-1)+\sum_{j=1}^{m}{s_{(j)}},$

i.e.

$\operatorname{reg}{\frac{S}{(d_{1},\ldots,d_{i})+J_{G}^{s}:d_{i+1}}}=2(s-1)+\sum_{j=1}^{m}{s_{(j)}}.$

Thus from Lemma 1.2(a) we obtain that
$\operatorname{reg}{{S}/{((d_{1},\ldots,d_{i})+J_{G}^{s}})}=2s+\sum_{j=1}^{m}{s_{(j)}}$,
for $s>2$. And for $s=2$ the statement follows from Lemma 4.1. This completes
the proof. ∎

######  Remark 4.1.

Note that the path graphs and star graphs are $\mathcal{T}_{m}$ graphs, for
suitable $s(i)$ and $m$. Using Theorem 4.1, we derive the following results.

  1. a)

[12, Theorem 3.5] Let $G=K_{1,n}$ be a star graph for $n\geq 3$. Then,
$\operatorname{reg}S/J_{G}^{s}=2s$ for all $s\geq 1$.

  2. b)

[12, Observation 3.2] Let $G=P_{n}$ be a path graph for $n\geq 2$. Then,
$\operatorname{reg}S/J_{G}^{s}=2s+n-3$ for all $s\geq 1$.

######  Remark 4.2.

Note that $T$-type graphs discussed in Shen and Zhu [21] are special case of
$\mathcal{T}_{m}$ graphs. Hence we obtain [21, Theorem 4.7] as a particular
case of Theorem 4.1.

Now, we focus on the regularity of powers of the binomial edge ideal of
$H_{m}$ graphs.

######  Theorem 4.2.

Let $G$ be a $\mathcal{H}_{m}$ graph on $[n+1]$. Let $d_{1},\ldots,d_{n}$ be a
sequence of edge binomials of $G$ such that $d_{1},\ldots,d_{n}$ forms a
$d$-sequence. Set $d_{0}=0\in S$. Then, for any $i=0,1,\ldots,n-1$, and for
any $m\geq 3$, and for all $s\geq 1$, we have

$\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{i})+J_{G}^{s}}=2s+\sum_{j=1}^{m+1}{s_{(j)}}+1.$

In particular,
$\operatorname{reg}{S}/{J_{G}^{s}}=2s+\sum_{j=1}^{m+1}{s_{(j)}}+1$, for all
$s\geq 1$.

###### Proof.

The proof is by induction on $s$. For $s=1$, the statement follows from Lemma
3.1. We may assume the statement for $s\geq 2$, and the assertion holds for
$s-1$ (take $d_{1},\ldots,d_{n}$ to be in the same order as in Theorem 2.2).
Now, the proof for the statement $s$ is by descending induction on $i$. For
$i=n-1$ the statement holds by Lemma 4.3. Next, assume that the assertion
holds for $i+1$. Consider the short exact sequence (4.5). By the induction
hypothesis on $i$ it follows that
$\operatorname{reg}{S}/{((d_{1},\ldots,d_{i+1})+J_{G}^{s})}=2s+\sum_{j=1}^{m+1}{s_{(j)}}+1$.
By the similar argument as in Theorem 4.1 we get
$((d_{1},\ldots,d_{i}):d_{i+1})+J_{G}^{s-1}=(d_{1},\ldots,d_{i})+J_{G}^{s-1}+J_{K_{n}}$,
for some $n<m$. Set $I=(d_{1},\ldots,d_{i})+J_{G}^{s-1}$ and $J=J_{K_{n}}$.
Consider the two cases:

Case 1: If $((d_{1},\ldots,d_{i}):d_{i+1})+J_{G}^{s-1}=I$, then from induction
hypothesis on $s$ it follows that
$\operatorname{reg}{S}/{I}=2(s-1)+\sum_{j=1}^{m+1}{s_{(j)}}+1$. Applying Lemma
1.2(a) to the short exact sequence (4.1) yields that

$\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{i})+J_{G}^{s}}=2s+\sum_{j=1}^{m+1}{s_{(j)}}+1.$

Case 2: If $((d_{1},\ldots,d_{i}):d_{i+1})+J_{G}^{s-1}=I+J$. Consider the
short exact sequence (4.6). By the induction hypothesis that
$\operatorname{reg}{S}/{I}=2(s-1)+\sum_{j=1}^{m+1}{s_{(j)}}+1$. Since $J$ is a
complete graph, from Remark 1.5(a) it follows that
$\operatorname{reg}{S}/{J}=1$.

Claim: $I\cap J=J*\langle x_{k_{0}},y_{k_{0}},J_{\mathcal{P}}\rangle$, where
$\mathcal{P}$ denotes an induced subgraph of $G$ on $V(G)\setminus k_{0}$ and
$k_{0}$ denote the center of $G$. The proof is similar to the proof of Claim
4.1 in Theorem 4.1. Now, From Theorem 3.1 we obtain that
$\operatorname{reg}S/(I\cap J)=2+\sum_{j=1}^{m+1}{s_{(j)}}$. Then from short
exact sequence (4.6) it follows that
$\operatorname{reg}{{S}/{(I+J)}}=2(s-1)+\sum_{j=1}^{m+1}{s_{(j)}}+1$, for all
$s\geq 2$. Thus, applying Lemma 1.2(a) to the short exact sequence (4.1)
yields

$\operatorname{reg}{\frac{S}{(d_{1},\ldots,d_{i})+J_{G}^{s}}}=2s+\sum_{j=1}^{m+1}{s_{(j)}}+1,\text{
for $s\geq 2$. }$

∎

To complete the proof of the Theorem 4.2, we need to prove the following
lemma.

######  Lemma 4.3.

Under the assumption in Theorem 4.2, and for all $s\geq 2$, we have

$\operatorname{reg}\frac{S}{(d_{1},\ldots,d_{n-1})+d_{n}^{s}}=2s+\sum_{j=1}^{m+1}{s_{(j)}}+1.$

###### Proof.

From Lemma 3.1 it follows that
$\operatorname{reg}{S}/{J_{G}}=2+\sum_{j=1}^{m+1}s_{(j)}+1$. By using Theorem
2.2 and Remark 1.4 one can obtain that
$\operatorname{reg}{S}/{(d_{1},\ldots,d_{n-1})}=2+\sum_{j=1}^{m+1}s_{(j)}$.
The rest of the proof is similar to the proof of Lemma 4.2. ∎

######  Remark 4.3.

Note that $H$-type graphs discussed in [21] is a particular case of graphs we
considered in Theorem 4.2. Hence we obtain [21, Theorem 4.7] as one specific
case of Theorem 4.2.

### Conclusion

The proof of Theorem 0.2 follows from Theorems 4.1 and 4.2.

We conclude the article with the following question:

######  Question 4.1.

Classify all finite simple graphs such that their edge binomials form a
$d$-sequence.

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# Modelling of transient interference phenomena in collinear laser
spectroscopy

Jovan Jovanović<EMAIL_ADDRESS>Theoretical Physics, Oxford
University, 1 Keble Road, Oxford OX1 3NP, United Kingdom  Ronald Fernando
Garcia Ruiz<EMAIL_ADDRESS>Massachusetts Institute of Technology,
Cambridge, MA 02139, USA

(August 27, 2024)

###### Abstract

Collinear laser spectroscopy of fast atomic beams has been established as one
of the main tools to perform precision experiments with atoms containing
short-lived nuclei. Although highly sensitive, the spectral resolution of
these techniques is typically limited to several MHz. Here, we study the use
of transient interference phenomena to potentially improve the experimental
resolution. Previous attempts to implement Ramsey-type measurements with fast
beams were limited by the lack of understanding of the observed line shapes.
By using a simple model, we provide a satisfactory description of the
previously observed line shapes, and propose alternative experimental schemes
to improve the resolution in fast ion-beam experiments.

##  I Introduction

Collinear laser spectroscopy techniques are considered as powerful tools to
perform systematic studies of atoms and molecules containing isotopes with
short-lived unstable nuclei [1, 2, 3, 4, 5, 6, 7]. As the Doppler width, as
seen along the axis of the motion of the atoms, is inversely proportional to
the kinetic energy of the atoms [1], narrow Doppler width measurements of
atomic spectra can be achieved by electrostatic acceleration. Thermal Doppler
widths of the order of GHz exhibited while at rest can be reduced to a few
tens of MHz by accelerating the ions to a few tens of keV. This resolution
allows for precise isotope shift measurements of atomic spectra, where changes
in the root-mean-square nuclear charge radii can be obtained along isotopic
chains. Moreover, the nuclear spins, magnetic dipole, and electric quadrupole
moments can be extracted from hyper fine structure measurements. These
techniques can now be applied to study isotopes produced in quantities lower
than just a few hundred ions per second [8, 9, 10], and with lifetimes as
short as a few milliseconds [11, 9].

In parallel to the experimental progress, the rapid increase of computer power
and the development of quantum many-body methods have enabled computations of
nuclear and atomic properties with unprecedented precision [12, 13, 14]. The
theoretical accuracy achieved in the last few years is now reaching the
precision of collinear laser spectroscopy experiments [15, 16, 10, 17].
Moreover, accurate calculations of higher order nuclear properties can now be
achieved [18, 19]. Hence improving the experimental precision, while
preserving high-efficiency, is becoming a major motivation of modern
experimental developments [19].

Conventional collinear laser spectroscopy techniques allows measurements of
atomic spectra with typical resolutions on the order of a few tens of MHz [20,
21]. Improving the resolution and precision of these techniques can provide
access to higher-order terms of the nuclear distribution, e.g. the fourth
power of the nuclear charge radius [18, 22, 23], and high-order moments of the
nuclear charge distribution [24]. Furthermore, precision isotope shift
measurements have been proposed as a viable route to constrain the existence
of possible new forces and particles[25]. This has motivated numerous
experimental and theoretical developments in high-precision isotope shift
spectroscopy using cold, trapped atoms and ions [22, 26, 27]. However, with a
few exceptions [28], trapping techniques are not in general applicable to
short-lived nuclei, which are produced in small quantities and with lifetimes
that can be as short as a fraction of a second [10, 11]. In this work we
explore the use of transient interference phenomena to study fast ion beams.
These techniques can be implemented to combine the high resolution of time-
dependent methods with the high sensitivity offered by collinear laser
spectroscopy.

##  II Transient interference in collinear laser spectroscopy

Transient interference phenomena in optical transitions have long been
recognized as a potent method to achieve high-resolution measurements well
below the natural linewidth of atomic transitions [29, 30, 31, 32]. In this
work we focus on the study of a near-resonance laser field interaction with an
ion travelling through different potential barriers. A scheme of the
experimental sequence is shown in Figure 1. The laser field is detuned from
the transition frequency in the ions’ rest frame. The velocity of the ion is
modified by passing through $n$ potential barriers, and tuned into resonance
at a potential $V_{0}$. Hence, each potential barriers is analogous to an
interaction region/period of a Ramsey-type measurement. When varying the
distance between two potentials, the population of the excited state
oscillates with a generalised Rabi frequency, from which the detuning, and
hence the resonance frequency, can be extracted.

This type of experiments were employed in collinear laser spectroscopy more
than four decades ago, using one potential [33], and two potential barriers
[34]. These results highlighted that the inherent high time resolution
achievable by using fast beams can be advantageous for spectroscopy techniques
[33, 35]. However, the Ramsey fringes that were experimentally observed [34,
33] exhibited relatively complex lineshapes that were not fully understood,
limiting the applicability of the techniques. In the following section we use
a simple model to study the interaction of a co-propagating laser and ion
beam, passing through $n$ potential barriers. This simple model provides a
satisfactory description of previous experiments using one potential [33], and
two potential barriers [34].

Figure 1: Scheme for Doppler switching of an ion beam passing through $n$
potential barriers. The electrical potentials change the ion energy, thus, the
frequency seen by the ions can be Doppler shifted around resonance (at
potential $V_{o}$).

##  III Two-level model

Considering a two-level atomic system coupled to a single electromagnetic (EM)
mode, we can account for the dominant physical processes: stimulated emission,
absorption and incoherent damping (e.g. spontaneous emission). A similar model
was used to study optical nutation in collinear fast beam experiments [36].
This system is described by a total Hamiltonian:

$\frac{1}{\hbar}\hat{H}(t)=\omega_{12}\ket{2}\bra{2}+\omega\hat{a}^{\dagger}\hat{a}+\frac{1}{\hbar}\sum_{\mu,\nu}\boldsymbol{F}\cdot\boldsymbol{D}_{\mu,\nu}i(\hat{a}e^{i\omega
t}-\hat{a}^{\dagger}e^{-i\omega t})\ket{\nu}\bra{\mu},$ (1)

where the two states are labelled as $\\{\mu,\nu\\}\in\\{1,2\\}$, $\hat{a}$ is
the annihilation operator for the photons in the mode coupled to the system
having frequency $\omega$. The energy difference between the two states is
denoted by $\hbar\omega_{12}$. The last term is the interaction Hamiltonian.
$\boldsymbol{D}_{\nu,\mu}$ represent the electric dipole matrix elements in
the basis of the two states, and
$\boldsymbol{F}=\sqrt{\frac{\omega}{2\epsilon_{0}\hbar
V}}\boldsymbol{\textbf{e}}$ characterises the electric field vector amplitude
of the mode.

This Hamiltonian acts on the states in the space:

$\ket{\Psi(t)}=\sum_{\mu}\ket{\xi_{\mu}(t)}\ket{\mu},$ (2)

where coefficients $\ket{\xi_{\mu}(t)}$ are vectors from the Hilbert space of
the EM field. The equation of motion that governs the time evolution of the
total density matrix of the system,

$i\hbar\frac{\partial}{\partial
t}\ket{\Psi(t)}\bra{\Psi(t)}=i\hbar\frac{\partial}{\partial
t}\hat{\rho}(t)=\commutator{\hat{H}(t)}{\hat{\rho}(t)},$ (3)

can be expressed in a rotating frame via the transformation
$\hat{U}_{\omega}(t)=(\ket{1}\bra{1}+e^{-i\omega t}\ket{2}\bra{2})\otimes
e^{-it\omega\hat{a}^{\dagger}\hat{a}}$:

$\hat{H}_{\omega}(t)=\hat{U}_{\omega}(t)\hat{H}(t)\hat{U}^{\dagger}_{\omega}(t)-i\hbar\frac{\partial\hat{U}_{\omega}(t)}{\partial
t}\hat{U}^{\dagger}_{\omega}(t).$ (4)

Working in this frame we can perform the usual steps in deriving the Lindblad-
type Master equation [37] for the reduced density matrix of the two-level
system:

$\rho_{S}(t)=\Tr_{EM}\rho(t),$ (5)

where the trace is taken over the degrees of freedom of the electric field.

One step we wish to highlight is where one integrates the Heisenberg’s
equation of motion (3) from some $t_{0}$ to $t$ and then plug that result back
into the same Heisenberg’s equation of motion to get:

$\frac{d}{dt}\rho_{S}(t)=\Tr_{EM}\big{\\{}-\frac{i}{\hbar}[H_{\omega}(t),\rho(t_{0})]\\\
-\frac{1}{\hbar^{2}}\int_{t_{0}}^{t}dt^{\prime}[H_{\omega}(t),[H_{\omega}(t^{\prime}),\rho(t^{\prime})]]\big{\\}}.$
(6)

This step is necessary in order for the following two approximations to yield
a non-trivial result. The first of the two approximation is the Born-
Oppenheimer approximation that states that the system is in a separable state:
$\rho(t)=\rho_{EM}(t)\otimes\rho_{S}(t)$. The second is the Markovian
approximation that states that the time dependence of the density operator can
only depend on the current state of the system and not it’s past:
$\rho(t^{\prime}),\rho(t_{0})\rightarrow\rho(t)$.

The reason for this highlight is to emphasise the origin of the physical
processes that happen when a two-level system is in contact with the EM
(laser) field. The first term, linear in $H_{\omega}$, is responsible for Rabi
oscillations characterised by the Rabi frequency
$\Omega_{Rabi}=\frac{1}{\hbar}\boldsymbol{E}_{0}\cdot\boldsymbol{D}_{1,2}$,
where $E_{0}$ is the expectation value of the amplitude of the oscillating
electric (laser) field. The second, quadratic, term is responsible for
stimulated emission and spontaneous decay (incoherent damping of Rabi
oscillations).

Performing the rest of the steps in deriving the Master equation and
neglecting terms oscillating more rapidly than the detuning
$\Delta=\omega_{12}-\omega$ (rotating-wave approximation), we arrive at the
optical Bloch equations for the matrix elements of the reduced density matrix
of the two-level system:

$\frac{d\rho_{1,1}}{dt}=-i\frac{\Omega_{Rabi}}{2}\rho_{1,2}+i\frac{\Omega_{Rabi}^{*}}{2}\rho_{2,1}+\Gamma\rho_{2,2},$

$\frac{d\rho_{1,2}}{dt}=-i\frac{\Omega_{Rabi}^{*}}{2}(\rho_{1,1}-\rho_{2,2})+(i\Delta-\frac{\Gamma}{2})\rho_{1,2},$

$\rho_{22}=1-\rho_{11}\quad\text{and}\quad\rho_{12}=\rho_{21}^{*},$ (7)

where $\Gamma=\frac{1}{\tau}$ is the inverse lifetime of the excited state.
The population of the excited state, $\rho_{2,2}$, can be probed by observing
the spontaneous emission intensity in the reference experiments or, for a
higher sensitivity, it could be measured through resonant ionization. [38]. As
$\commutator{\hat{U}_{\omega}(t)}{\hat{U}_{\omega^{\prime}}(t)}=0$, these
equations can be used in the case of time-dependent detuning by replacing
$\Delta$ with $\Delta_{D}(t)$. The spatial dependence of the electromagnetic
field can be introduced as a phase factor on the correlation coefficients.
Thus, it does not contribute to the observed populations [36]. Ramsey
protocols can be studied by using a time-dependent Rabi frequency in the
equations above.

##  IV Doppler switching through different potential barriers

The population of the excited state of a two-level system interacting with a
laser field near resonance was studied by using different potential barriers
(see Figure 1). The equations described in section III were numerically
integrated for different system configurations in order to study their
performance in a Ramsey-like experiment. The value of $\Gamma$ was obtained
from the inverse sum of Einstein’s A coefficients for a relevant state, which
are taken from the NIST database [39]. Firstly, we test the model description
with existing experimental results. A satisfactory description of previous
observations allows us to study and propose alternative spectroscopy
protocols.

###  IV.1 One potential barrier

Ramsey fringes in collinear laser spectroscopy have been experimentally
observed by using one potential barrier [34, 33]. The experimental scheme is
shown in Figure 2. Using a fixed potential, the laser was scanned around the
Doppler-shifted resonant frequency. The population of the excited state was
probed by detecting fluorescent photons in a detector placed after the
potential barrier. The experimental results from Ref. [34, 33] are compared
with our numerical results in Figure 3.

The radiation intensity ($\sim E_{0}$) was obtained from the laser power and
ion beam diameter reported in [34]. The dipole matrix element, $ea_{B}$, was
adjusted to give an order of magnitude estimate for Rabi frequency of 100 MHz
(this is tunable in any setup). The geometry of the experimental schemes was
taken as reported in Refs. [34, 33]. Laser detuning with respect to the probed
transition was chosen to fit the experimental data and illustrate certain
features thereof. The results are shown for the 455 nm transition line of Ba
II ($6s^{2}{S}_{1/2}$ $\rightarrow$ $6p^{2}{P}_{3/2}$). The Einstein’s A
coefficient ($\sim 10^{8}s^{-1}$ MHz) for this transition was taken from [40].
The electric field produced by the potential barrier for this geometry was
simulated using the software package COMSOL. The simulated electrostatic
potential was used to provide a realistic estimation of the velocity
distribution of the ions.

Figure 2: Scheme of Doppler switching with one potential barrier. The Doppler-
shifted frequency as a function of time of flight is shown in blue . The
resonant laser detuning needed to drive the transition is shown in orange .
The frequency is shown relative to the laser frequency as seen in the ions’
rest frame in the regions free from the electrostatic field. The green region
shows interrogation time over which the excited population is integrated.
Figure 3: Excited state population of an ion passing through one potential
barrier. The amplitude of the potential barrier was fixed, and the laser
frequency was scanned around the Doppler-shifted resonance frequency. The
experimental results from Ref. [34] are compared with our theoretical results.
The curves on the log-linear plot are offset on purpose, the overall
multiplicative normalisation factor is setup dependent, hence, in this paper,
arbitrary.

As seen in Figure 3, a good agreement between theory and experiment is
observed. Previous analysis neglected the lifetime of the excited state, the
linewidth of the laser, and did not employ a realistic velocity distribution
[34, 33]. In our current model, the lifetime was directly accounted for in the
equation of motion and the laser linewidth was included through convolution
with a Lorentzian lineshape. This approach is valid when the time of flight is
comparable to the Rabi period, meaning that during that time the ion interacts
with only one mode of the EM field. The linewidth of the laser is relevant
when we have an ensemble of ions interacting with different modes of the laser
light. Hence, we add contributions from different modes, i.e. by convolution
of the frequency scan with the lineshape. Thermal effects were neglected. This
scheme provides a test of the simple theoretical description, but is not
suitable for Ramsey spectroscopy since scanning through interaction times by
changing the velocity of the atoms would in-turn change the detuning.

###  IV.2 Two potential barriers

In this section we study the population of the excited state of a two-level
system when it is passed through two different potential barriers. The
experimental scheme is shown in Figure 4. Experimental results for this
configuration were presented in [34], but the observed lineshapes were not
satisfactorily explained. The parameters of the model were obtained as
explained in the previous section. Similarly, the electric potentials of the
geometry stated in Ref. [34] was simulated in COMSOL, and the simulated
electric fields were used to provide a realistic distribution of the ion beam
velocities.

Figure 4:  Doppler switching with two potential barriers: Dependence of
Doppler-shifted frequency on the time of flight. The resonant frequency as
referenced from the Doppler-shifted laser frequency outside the electrostatic
field is shown (blue solid, blue dashed). The green region shows interrogation
time over which the excited population is integrated. The detuning over time
(blue dashed) was modelled by simulating the electrostatic potential with the
geometry and potentials of the setup reported in [34]. A sum of two Gaussians
(blue solid) was used as an approximation to simplify our simulations.

The experimental and theoretical results are presented in Figure 5. A good
agreement between the experimental lineshapes [34] and our simulations is
observed. Previous attempts to explain these lineshapes did not provide a
complete description [34]. We noticed that in addition to the model
differences, the simulated electric fields were significantly different to the
assumed functional forms used in Ref. [34, 33].

In the experiment, the laser detuning was being scanned. This is illustrated
in Figure 4 where the position of the orange line, i.e. the value of the laser
detuning relative to the resonance, can be varied. This is equivalent to
changing the position along the ions’ path where the laser frequency is on
resonance with the ionic transition. These points correspond to the
intersections of the blue and orange lines in Figure 4. On the same figure we
see that depending on the detuning there can exist two or four positions at
which the ion is on resonance with the laser field. In the case where there
are four points, the ions effectively see two potential barriers. This double
switch case exhibits Ramsey oscillations at the fixed frequency corresponding
to the separation of the two peaks. When the detuning (orange curve) is lower
than some critical value (around 800 MHz in this case), only two intersections
occur between the blue and orange lines. This case is similar to a single
potential barrier discussed in the previous section. The ions do not "see" the
dip of the frequency in the centre since it is detuned from resonance. These
two different features are present in the experimental data [34] shown in
Figure 5.

Figure 5: Excited state population of an ion passing through two potential
barriers. The amplitude of the potential barriers were fixed, and the laser
frequency was scanned around the Doppler-shifted resonance frequency. The
experimental results from Ref. [34] are compared with our simulations. The
curves on the log-linear plot are offset on purpose, the overall
multiplicative normalisation factor is setup dependent, hence, in this paper,
arbitrary.

It is important to note that in both single and double barrier cases, the main
Rabi-type peaks at no detuning in Figures 3 and 5, are not broken up by Ramsey
fringes. Hence, these protocols offer no improvement on the precision of
determining the resonance position, as it is seen in the stationary Ramsey
spectroscopy.

We will offer a protocol that can improve the precision in the following
sections.

##  V Time Scanning with two potential barriers

The good agreement of our simple model for one and two potential barriers,
motivates us to explore the use of transient field phenomena to perform future
precision studies. Here we propose a measurement protocol to implement Ramsey
spectroscopy, using two Gaussian potential barriers separated by a variable
distance. In this configuration, the population of the excited state can be
measured as a function of the distance, and the frequency detuning can be
obtained from the Fourier transform of the observed signal [41].

Figure 6: Doppler switching with two potential barriers of variable distance.
The dependence of Doppler-shifted frequency as a function of time of flight is
shown with the blue curves. The separation between two voltage pulses can be
varied. The resonant frequency as referenced from the Doppler-shifted laser
frequency outside the electrostatic field is marked with the orange curve. The
green region shows the interrogation time over which the population of the
excited state is integrated.

The protocol for the proposed experimental scheme is shown in Figure 6. The
distance between the two potential barriers (blue peaks) can be varied, and
the population of the excited state can be measured as a function of the time-
of-flight separation (Ramsey oscillations). This scheme can be implemented as
a layered structure of concentric rings stacked on top of one another, similar
to the setup presented in [34] but at smaller scale. Such an electrode array
can be manufactured by means of 3D printing [42, 43]. Assuming a period of the
layered structure of about 0.5 mm over a 0.5 m span, a resolution of 1 kHz and
bandwidth of the 1 GHz can be achieved for an ion mass of $A=100$ accelerated
to 20 keV.

Figure 7:  Results from the time domain scan: a) Ramsey oscillations seen in
the fluorescent signal versus time between pulses, the points shown are taken
from an arbitrary narrow window of the total 1000 ns simulated range. The
oscillations seen are in agreement with the generalised Rabi frequency of our
setup and the decay rate is half of the natural linewidth. The experimental
points and the decaying exponential fit are shown in blue and orange
respectively; b) Fourier transform of the Ramsey oscillations for different
natural linewidths. The dashed line shows the Lorentzian fit; c) resolution
obtained from the fit of the Ramsey fringes to Lorentzian shapes. The width of
the Lorentizan shape, Ramsey linewidth, is shown as a function of the natural
linewidth.

The simulations of the proposed scheme were performed using the following
parameters: a pulse length, the time ions experience the electrostatic field
of the barrier, of 0.1 ns; a pulse height, the maximal Doppler shift
experienced by the atoms as they slow down while crossing the barrier, of 1880
MHz; a detuning of 900 MHz; a natural linewidth of 1 MHz; a Rabi frequency of
600 Mrad/s; a range of separation from 0 up to 1000 ns. The population of the
excited state was obtained as an average over 200 ns, and assumed to be
measured 250 ns after the last potential barrier. The Ramsey oscillations
simulated for the population of the excited state are shown in Figure 7a. The
Fourier transformed of the signal is shown in Figure 7b. This procedure
measures the generalised Rabi frequency, $\Omega_{GEN}$, which is related to
the frequency detuning from resonance, $\Delta$, and the Rabi frequency
$\Omega_{Rabi}$

$\Omega_{GEN}^{2}=\Delta^{2}+\Omega_{Rabi}^{2}.$ (8)

High laser powers can cause a detuning from the laser frequency due to the AC
Stark-like shift, without introducing power broadening, which can be a major
disadvantage in Rabi-type measurement.

We have not modelled noise in this study and the parameters we chose were
arbitrary, however, the following two facts are relevant for the actual
experimental setup: i. the area under the pulse in the detuning profile
measured in the time-of-flight domain determines the visibility of the fringes
(the AC amplitude divided by the DC level of the fluorescence); ii. the Rabi
frequency determines the magnitude of the overall signal (both DC and AC).
Hence the actual signal-to-noise ratio heavily depends on the details of a
particular experiment.

The lineshape of the Ramsey oscillations was studied using the same model
parameters and scanning from 200 ns to 4200 ns. The simulations were performed
assuming different natural linewidths. The result of this analysis is
represented in Figure 7. A Lorentzian curve fits the lineshape well, with a
centroid that is within 10kHz (for the narrowest 1kHz) of the expected value.
Fitting the time domain can offer sub-kHz agreement across all linewidths. As
can be seen from equation (7), Ramsey lines are twice as narrow than the
equivalent lineshapes obtained for Rabi-type measurements. The coherences’,
which are responsible for the oscillations, decay time is twice as long as
populations’ decay time.

The AC Stark-like shift can be corrected by measuring the frequency at
different laser powers. This procedure is illustrated in Figure 8. Frequency
measurement with different powers of the probing laser, $P$, can be expressed
as $P=\chi^{-1}\Omega_{Rabi}^{2}$, with $\chi$ an experimentally dependent
proportionality constant that can be obtained by fitting the function
$f(P)=\sqrt{\Delta^{2}+\chi P}$ to the experimental results.

Figure 8: Relationship between the Ramsey frequency and Rabi frequency. The
solid line representing the expected dependence.

##  VI Summary and outlook

Motivated by the potential use of transient phenomena to improve the spectral
resolution in collinear laser spectroscopy, we present a simple description of
a co-propagating laser interacting with an ion beam travelling through
different potential barriers. This model provides a good description of
previous experimental results that were not fully understood. A experimental
scheme was proposed with the aim of implementing Ramsey-type measurements with
fast ion beams for future precision studies. The proposed measurement protocol
allows for varying the time-of-flight separation between potential barriers.

Two main advantages of the proposed transient interference protocol over the
usual Rabi-type collinear spectroscopy can be highlighted: i. the detuning is
measured by observing the coherences of the density matrix, which results in
half of the linewidth; ii. power broadening is eliminated in favour of an AC
Stark effect, which can be empirically obtained.

Our results neglected the thermal motion of the ions along the beam direction.
Including this effect would be important for an accurate description of
precision experiments.

###### Acknowledgements.

This work is supported by Department of Energy, Office of Science, Office of
Nuclear Physics, under Award numbers DE-SC0021176 and DE-SC0021179. JJ is
partly supported by the Oxford-ShanghaiTech collaboration agreement. We thank
to Silviu Udrescu, Adam Vernon, and Shane Wilkins for fruitful comments and
suggestions.

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