# A note on Fujimoto’s uniqueness theorem with $(2n+3)$ hyperplanes

Kai Zhou 111E-mail address<EMAIL_ADDRESS>
###### Abstract

Hirotaka Fujimoto proved in [Nagoya Math. J., 1976(64): 117–147] and [Nagoya
Math. J., 1978(71): 13–24] a uniqueness theorem for algebraically non-
degenerate meromorphic maps into $\mathbb{P}^{n}(\mathbb{C})$ sharing $(2n+3)$
hyperplanes in general position. The proof of Lemma 3.6 in [Nagoya Math. J.,
1976(64): 117–147] is found to contain a mistake. This mistake can be
corrected easily from a new point of view. This note explains the idea of the
correction and provides a complete proof. ††2010 Mathematics Subject
Classification. 32H30, 20K15.

##  1 Introduction

In the 1970s, Hirotaka Fujimoto [1, 2, 3] proved several uniqueness theorems
for meromorphic maps into a complex projective space, which are
generalizations of the uniqueness theorems of meromorphic functions obtained
in [5] and [4]. For instance, Fujimoto proved in [2] and [3] the following
uniqueness theorem.

######  Theorem 1.1.

Let $f$ and $g$ be two meromorphic maps of $\mathbb{C}^{m}$ into the
$n$-dimensional complex projective space $\mathbb{P}^{n}(\mathbb{C})$ such
that $f(\mathbb{C}^{m})\not\subseteq H_{j}$, $g(\mathbb{C}^{m})\not\subseteq
H_{j}$ and $f^{*}(H_{j})=g^{*}(H_{j})\,(1\leq j\leq 2n+3)$ for $(2n+3)$
hyperplanes $H_{j}$ in $\mathbb{P}^{n}(\mathbb{C})$ in general position. If
$g$ is algebraically non-degenerate, then $f=g.$

One of the key ingredients of Fujimoto’s proof of the above theorem is the
Lemma 3.6 in [2]. Unfortunately, a mistake is found in Fujimoto’s proof of the
Lemma 3.6 in [2]. In this note, I shall show this mistake and explain the idea
of correcting it. A complete proof of the Lemma 3.6 in [2] will be given in
Section 2.

To state Fujimoto’s Lemma 3.6 in [2], we need some preparations.

Let $G$ be a torsion-free abelian group and let
$A=(\alpha_{1},\dots,\alpha_{q})$ be a $q$-tuple of elements in $G.$ Denote by
$\langle\alpha_{1},\dots,\alpha_{q}\rangle$ the subgroup of $G$ generated by
$\alpha_{1},\dots,\alpha_{q}.$ Since $G$ is torsion-free, the subgroup
$\langle\alpha_{1},\dots,\alpha_{q}\rangle$ is free and is of finite rank. We
denote by rank$\\{\alpha_{1},\dots,\alpha_{q}\\}$ the rank of
$\langle\alpha_{1},\dots,\alpha_{q}\rangle.$

The following notion is introduced by Fujimoto.

######  Definition 1.2.

Let $q\geq r>s\geq 1$ be positive integers. Let $(G,\cdot)$ be a torsion-free
abelian group. A $q$-tuple $A=(\alpha_{1},\dots,\alpha_{q})$ of elements in
$G$ is said to _have the property_ $(P_{r,s})$ if arbitrarily chosen $r$
elements $\alpha_{l(1)},\dots,\alpha_{l(r)}$ in $A$ ($1\leq
l(1)<\dots<l(r)\leq q$) satisfy the condition that, for any $s$ distinct
indices $i_{1},\dots,i_{s}\in\\{1,\dots,r\\},$ there exist distinct indices
$j_{1},\dots,j_{s}\in\\{1,\dots,r\\}$ with
$\\{j_{1},\dots,j_{s}\\}\neq\\{i_{1},\dots,i_{s}\\}$ such that

$\alpha_{l(i_{1})}\cdot\alpha_{l(i_{2})}\cdots\alpha_{l(i_{s})}=\alpha_{l(j_{1})}\cdot\alpha_{l(j_{2})}\cdots\alpha_{l(j_{s})}.$

We also need the following notation.

######  Notation.

For elements
$\alpha_{1},\alpha_{2},\dots,\alpha_{q},\tilde{\alpha}_{1},\tilde{\alpha}_{2},\dots,\tilde{\alpha}_{q}$
in an abelian group $(G,\cdot),$ we write

$\alpha_{1}:\alpha_{2}:\dots:\alpha_{q}=\tilde{\alpha}_{1}:\tilde{\alpha}_{2}:\dots:\tilde{\alpha}_{q},$

if there is an element $\beta\in G$ such that
$\alpha_{i}=\beta\tilde{\alpha}_{i}$ for any $1\leq i\leq q.$

Now the Lemma 3.6 in [2] is stated as follows.

######  Lemma 1.3.

Let $s$ and $q$ be positive integers with $2\leq s<q\leq 2s.$ Let $(G,\cdot)$
be a torsion-free abelian group. Let $A=(\alpha_{1},\dots,\alpha_{q})$ be a
$q$-tuple of elements in $G$ that has the property $(P_{q,s}),$ and assume
that at least one $\alpha_{i}$ equals the unit element $1$ of $G.$ Then

  1. (i)

rank$\\{\alpha_{1},\dots,\alpha_{q}\\}=:t\leq s-1;$

  2. (ii)

if $t=s-1,$ then $q=2s$ and there is a basis
$\\{\beta_{1},\dots,\beta_{s-1}\\}$ of
$\langle\alpha_{1},\dots,\alpha_{q}\rangle$ such that the $\alpha_{i}$ are
represented, after a suitable change of indices, as one of the following two
types:

     * (A)

$s$ is odd and

$\alpha_{1}:\alpha_{2}:\dots:\alpha_{2s}=1:1:\beta_{1}:\beta_{1}:\beta_{2}:\beta_{2}:\dots:\beta_{s-1}:\beta_{s-1};$

     * (B)

$\alpha_{1}:\alpha_{2}:\dots:\alpha_{2s}=1:1:\dots:1:\beta_{1}:\dots:\beta_{s-1}:(\beta_{1}\cdots\beta_{a_{1}})^{-1}:(\beta_{a_{1}+1}\cdots\beta_{a_{2}})^{-1}:\dots:(\beta_{a_{k-1}+1}\cdots\beta_{a_{k}})^{-1},$

where $0\leq k\leq s-1$, $1\leq a_{1}<a_{2}<\dots<a_{k}\leq s-1,$ and the unit
element $1$ appears $(s+1-k)$ times in the right hand side.

Fujimoto’s proof of the above lemma essentially uses the assertion that, for
any $q$-tuple $A=(\alpha_{1},\dots,\alpha_{q})$ of elements in a torsion-free
abelian group $(G,+),$ there exists a basis $\\{\eta_{1},\dots,\eta_{t}\\}$ of
$\langle\alpha_{1},\dots,\alpha_{q}\rangle$ with the following property: there
are distinct indices $i_{1},\dots,i_{t}\in\\{1,\dots,q\\}$ and nonzero
integers $l_{1},\dots,l_{t}$ such that

$\alpha_{i_{\tau}}=l_{\tau}\eta_{\tau},\quad 1\leq\tau\leq t.$

Such a basis $\\{\eta_{1},\dots,\eta_{t}\\}$ is called by Fujimoto an adequate
basis for $A.$ Unfortunately, adequate basis does not exist in general. This
is illustrated by the following example.

######  Example 1.4.

Consider the free abelian group $(\mathbb{Z}^{3},+).$ Let

$\alpha_{1}=(1,0,0),\quad\alpha_{2}=(1,1,0),\quad\alpha_{3}=(1,2,2),\quad\alpha_{4}=(1,2,5)$

be four elements in $\mathbb{Z}^{3}.$

Since

$(0,1,0)=\alpha_{2}-\alpha_{1}\quad\text{and}\quad(0,0,1)=-2\alpha_{3}+\alpha_{4}+2\alpha_{2}-\alpha_{1},$

we see that

$\langle\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\rangle=\mathbb{Z}^{3}.$

So, if there exists an adequate basis for
$(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}),$ then one sees easily that
there are distinct indices $i,j,k\in\\{1,2,3,4\\}$ such that
$\\{\alpha_{i},\alpha_{j},\alpha_{k}\\}$ is a basis of $\mathbb{Z}^{3}.$
However, one verifies easily that $\\{\alpha_{i},\alpha_{j},\alpha_{k}\\}$ is
not a basis of $\mathbb{Z}^{3}$ for any three distinct indices
$i,j,k\in\\{1,2,3,4\\}.$ This shows that adequate basis does not exist for
$(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}).$

Fujimoto’s proof of Lemma 1.3 can be corrected easily from a new point of
view. Roughly speaking, we take an “adequate basis” in
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q}$ and
regard
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q}$ as
the ambient space. I shall explain more.

Let $(G,+)$ be a torsion-free abelian group and let
$A=(\alpha_{1},\dots,\alpha_{q})$ be a $q$-tuple of elements in $G.$ We regard
the free abelian group $\langle\alpha_{1},\dots,\alpha_{q}\rangle$ as a
subgroup of
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q}$
which can be naturally regarded as a $\mathbb{Q}$-vector space. Because the
$\mathbb{Q}$-vector space
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q}$ is
generated by $\alpha_{1},\dots,\alpha_{q},$ there exist distinct indices
$i_{1},\dots,i_{t}\in\\{1,\dots,q\\}$ with $t=\mbox{\rm
rank}\\{\alpha_{1},\dots,\alpha_{q}\\}$ such that
$\\{\alpha_{i_{1}},\dots,\alpha_{i_{t}}\\}$ is a basis of the
$\mathbb{Q}$-vector space
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q}.$
Then we see that there exist positive integers $l_{1},\dots,l_{t}$ such that,
for every $j\in\\{1,\dots,q\\}\setminus\\{i_{1},\dots,i_{t}\\},$

$\alpha_{j}=l(j,1)\cdot\frac{1}{l_{1}}\alpha_{i_{1}}+\dots+l(j,t)\cdot\frac{1}{l_{t}}\alpha_{i_{t}}$

for suitable integers $l(j,1),\dots,l(j,t).$

The above discussion shows the following.

######  Observation 1.5.

Let $(G,+)$ be a torsion-free abelian group and let
$A=(\alpha_{1},\dots,\alpha_{q})$ be a $q$-tuple of elements in $G.$ We regard
the free abelian group $\langle\alpha_{1},\dots,\alpha_{q}\rangle$ as a
subgroup of
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q}.$
There exist distinct indices $i_{1},\dots,i_{t}\in\\{1,\dots,q\\}$ with
$t=\mbox{\rm rank}\\{\alpha_{1},\dots,\alpha_{q}\\},$ positive integers
$l_{1},\dots,l_{t}$ and a basis $\\{\eta_{1},\dots,\eta_{t}\\}$ of the
$\mathbb{Q}$-vector space
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q},$
such that

$\displaystyle\alpha_{i_{\tau}}$ $\displaystyle=l_{\tau}\eta_{\tau},\quad
1\leq\tau\leq t,$ $\displaystyle\alpha_{j}$
$\displaystyle=l(j,1)\eta_{1}+\dots+l(j,t)\eta_{t},\quad
j\in\\{1,\dots,q\\}\setminus\\{i_{1},\dots,i_{t}\\},$

where $l(j,1),\dots,l(j,t)$ are integers.

A complete proof of Lemma 1.3 will be given in Section 2. In the proof, we
take a basis $\\{\eta_{1},\dots,\eta_{t}\\}$ as in Observation 1.5, and we
regard the $\alpha_{i}$ as elements in
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q}$
whenever it is needed. The proof basically follows Fujimoto’s original
strategy. But simplifications are made in some steps of Fujimoto’s strategy.

We shall give another two observations for later use.

######  Observation 1.6.

Let $(\alpha_{1},\dots,\alpha_{q})$ and
$(\tilde{\alpha}_{1},\dots,\tilde{\alpha}_{q})$ be two $q$-tuples of elements
in a torsion-free abelian group $(G,\cdot).$ Assume that there exist indices
$i_{0}$ and $j_{0}$ such that $\alpha_{i_{0}}=\tilde{\alpha}_{j_{0}}=1,$ and
there is an element $\beta\in G$ such that
$\alpha_{i}=\beta\tilde{\alpha}_{i}$ for any $1\leq i\leq q.$ Then

$\langle\alpha_{1},\dots,\alpha_{q}\rangle=\langle\tilde{\alpha}_{1},\dots,\tilde{\alpha}_{q}\rangle.$

######  Observation 1.7.

Let $x_{1},\dots,x_{t+k}$ be elements in a torsion-free abelian group
$(G,\cdot),$ which can be represented as follows:

$\displaystyle(x_{1},\dots,x_{t},x_{t+1},\dots,x_{t+k})$
$\displaystyle\qquad=\big{(}\beta_{1},\dots,\beta_{t},(\beta_{1}\cdots\beta_{a_{1}})^{-1},(\beta_{a_{1}+1}\cdots\beta_{a_{2}})^{-1},\dots,(\beta_{a_{k-1}+1}\cdots\beta_{a_{k}})^{-1}\big{)},$

where $t\geq 1$, $0\leq k\leq t$, $1\leq a_{1}<\dots<a_{k}\leq t,$ and
$\beta_{1},\dots,\beta_{t}$ are multiplicatively independent elements in $G.$
For any $t$ multiplicatively independent elements $x_{i_{1}},\dots,x_{i_{t}}$
among $x_{i}$’s, by writing
$\\{1,\dots,t+k\\}\setminus\\{i_{1},\dots,i_{t}\\}=\\{j_{1},\dots,j_{k}\\},$
we can verify that

$(x_{i_{1}},\dots,x_{i_{t}},x_{j_{1}},\dots,x_{j_{k}})=\big{(}x_{i_{1}},\dots,x_{i_{t}},\big{(}\prod_{i\in\Lambda_{1}}x_{i}\big{)}^{-1},\dots,\big{(}\prod_{i\in\Lambda_{k}}x_{i}\big{)}^{-1}\big{)},$

where $\Lambda_{1},\dots,\Lambda_{k}$ are nonempty subsets of
$\\{i_{1},\dots,i_{t}\\}$ with $\Lambda_{i}\cap\Lambda_{j}=\emptyset$ for any
$1\leq i<j\leq k.$

We also recall the following result of Fujimoto.

######  Lemma 1.8 (see Lemma 2.6 in [1]).

Let $G$ be a torsion-free abelian group. Let $A=(\alpha_{1},\dots,\alpha_{q})$
be a $q$-tuple of elements in $G$ that has the property $(P_{r,s}),$ where
$q\geq r>s\geq 1.$ Then there exist two distinct indices
$i,j\in\\{1,\dots,q\\}$ such that $\alpha_{i}=\alpha_{j}.$

The following notation will be used in Section 2.

######  Notation.

For a set $S,$ we denote by $\\#S$ the cardinality of $S.$

##  2 Proof of Lemma 1.3

Lemma 1.3 will be proved by the induction on $s.$

Consider first the case of $s=2.$ In this case, $q=3$ or $q=4.$

When $q=3,$ we claim that $\alpha_{1}=\alpha_{2}=\alpha_{3}.$ Assume this is
not true. Without loss of generality, we may assume that
$\alpha_{1}\neq\alpha_{2}.$ Because
$\alpha_{1}\alpha_{3}\neq\alpha_{2}\alpha_{3},$ by assumption, we see
$\alpha_{1}\alpha_{3}=\alpha_{1}\alpha_{2},$ which implies
$\alpha_{3}=\alpha_{2}.$ Similarly, we get $\alpha_{3}=\alpha_{1}.$ Then
$\alpha_{1}=\alpha_{2},$ which gives a contradiction. Thus
$\alpha_{1}=\alpha_{2}=\alpha_{3}=1$ and
rank$\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}=0.$ This shows the conclusions of
Lemma 1.3 hold in this case.

Now assume $q=4.$ It follows from Lemma 1.8 that there are distinct indices
$i_{0},j_{0}\in\\{1,\dots,4\\}$ such that $\alpha_{i_{0}}=\alpha_{j_{0}}.$
Then by Observation 1.6 and considering the new 4-tuple
$(\alpha_{i_{0}}^{-1}\alpha_{1},\dots,\alpha_{i_{0}}^{-1}\alpha_{4}),$ we may
assume, after a suitable change of indices, that $\alpha_{1}=\alpha_{2}=1.$ By
assumption, there are distinct indices $i,j\in\\{1,\dots,4\\}$ with
$\\{i,j\\}\neq\\{1,2\\}$ such that
$\alpha_{1}\alpha_{2}=\alpha_{i}\alpha_{j}.$ If $\\{i,j\\}\neq\\{3,4\\},$ then
we see easily that $\alpha_{3}=1$ or $\alpha_{4}=1.$ If $\\{i,j\\}=\\{3,4\\},$
then $\alpha_{3}=\alpha_{4}^{-1}.$ So at least one of the following three
cases occurs:

  * •

$(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})=(1,1,1,\beta);$

  * •

$(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})=(1,1,\beta,1);$

  * •

$(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})=(1,1,\beta,\beta^{-1}).$

It follows that rank$\\{\alpha_{1},\dots,\alpha_{4}\\}\leq 1,$ and if
rank$\\{\alpha_{1},\dots,\alpha_{4}\\}=1,$ then
$(\alpha_{1},\dots,\alpha_{4})$ is of the type (B).

Now Lemma 1.3 is proved in the case of $s=2.$

Next we assume that $s\geq 3$ and assume the conclusions hold when $s$ in
Lemma 1.3 is replaced by a number $\leq s-1.$

If rank$\\{\alpha_{1},\dots,\alpha_{q}\\}=:t<s-1,$ then the conclusions of
Lemma 1.3 hold trivially. Therefore, we assume in the following that $t\geq
s-1.$

Set

$M_{0}:=\big{\\{}i\in\\{1,\dots,q\\}\,|\,\alpha_{i}=1\big{\\}}\quad\mbox{and}\quad
m_{0}:=\\#M_{0}.$

By Lemma 1.8 and Observation 1.6, we may assume that $m_{0}\geq 2.$

We may regard $\langle\alpha_{1},\dots,\alpha_{q}\rangle$ as a subgroup of
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q}.$ By
Observation 1.5, there exist distinct indices
$u_{1},\dots,u_{t}\in\\{1,\dots,q\\},$ positive integers $l_{1},\dots,l_{t},$
and a basis $\\{\eta_{1},\dots,\eta_{t}\\}$ of the $\mathbb{Q}$-vector space
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q},$
such that, the $\alpha_{i},$ as elements in
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q},$ are
represented as follows:

$\alpha_{i}=\eta_{1}^{l(i,1)}\cdot\cdots\cdot\eta_{t}^{l(i,t)},\quad 1\leq
i\leq q,$ (2.1)

where $l(i,1),\dots,l(i,t)$ are integers and, for each $1\leq\tau\leq t,$

$l(u_{\tau},\tau)=l_{\tau}(>0)\quad\mbox{and}\quad
l(u_{\tau},\tau^{\prime})=0\,\,\mbox{for any}\,\,\tau^{\prime}\neq\tau.$

The rest of the proof is divided into the following two cases:

Case $(\alpha)$

For every $1\leq\tau\leq t,$ the $q$ integers $l(1,\tau),\dots,l(q,\tau)$ are
all non-negative;

Case $(\beta)$

There is some $\tau\in\\{1,\dots,t\\}$ such that $l(i,\tau)<0$ for some
$i\in\\{1,\dots,q\\}.$

###  2.1 The proof for the Case $(\alpha)$

For each $1\leq\tau\leq t,$ we put

$M_{\tau}:=\big{\\{}i\in\\{1,\dots,q\\}\,|\,l(i,\tau)>0,\,l(i,\tau+1)=\dots=l(i,t)=0\big{\\}}$

and

$m_{\tau}:=\\#M_{\tau}.$

Every $M_{\tau}$ is nonempty, since $u_{\tau}\in M_{\tau}.$

We note here that, for any $i\in M_{0}$, $l(i,1)=l(i,2)=\dots=l(i,t)=0.$ So
these $M_{\tau}$ give the following partition of $\\{1,\dots,q\\}$:

$\\{1,\dots,q\\}=M_{0}\cup M_{1}\cup\cdots\cup M_{t}.$

######  Claim 2.2.

For any subset $\\{\tau_{1},\dots,\tau_{p}\\}$ of $\\{0,1,\dots,t\\}$ with
$\tau_{1}<\tau_{2}<\dots<\tau_{p},$

$m_{\tau_{1}}+\dots+m_{\tau_{p}}\neq s.$

###### Proof.

Assume on the contrary that $m_{\tau_{1}}+\dots+m_{\tau_{p}}=s$ for some
$\tau_{1},\dots,\tau_{p}$ with $0\leq\tau_{1}<\dots<\tau_{p}\leq t.$ Write

$M_{\tau_{1}}\cup\cdots\cup M_{\tau_{p}}=\\{i_{1},\dots,i_{s}\\}=:I.$

Consider the $s$ elements $\alpha_{i_{1}},\dots,\alpha_{i_{s}}.$ By
assumption, there exist distinct indices $j_{1},\dots,j_{s}\in\\{1,\dots,q\\}$
with $\\{j_{1},\dots,j_{s}\\}=:J\neq I,$ such that

$\alpha_{i_{1}}\cdot\dots\cdot\alpha_{i_{s}}=\alpha_{j_{1}}\cdot\dots\cdot\alpha_{j_{s}}.$
(2.3)

We regard the $\alpha_{i}$ as elements in
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q}$ and
substitute (2.1) into the above equation. Then we get

$\eta_{1}^{l(i_{1},1)+\dots+l(i_{s},1)}\cdots\eta_{t}^{l(i_{1},t)+\dots+l(i_{s},t)}=\eta_{1}^{l(j_{1},1)+\dots+l(j_{s},1)}\cdots\eta_{t}^{l(j_{1},t)+\dots+l(j_{s},t)}.$
(2.4)

For each $1\leq\tau\leq t,$ the exponent of $\eta_{\tau}$ in the left hand
side of (2.4) is necessarily equal to that in the right hand side.

We consider the following three cases.

$\bullet\quad\tau_{1}\geq 1.$

Let $\tau\in\\{\tau_{p}+1,\dots,t\\}.$ Since $I=M_{\tau_{1}}\cup\cdots\cup
M_{\tau_{p}},$ the exponent of $\eta_{\tau}$ in the left hand side of (2.4) is
zero, and thus $J\cap M_{\tau}=\emptyset.$ For, if $J\cap
M_{\tau}\neq\emptyset,$ then the exponent of $\eta_{\tau}$ in the right hand
side of (2.4) is strictly greater than zero, and this gives a contradiction.

Then by observing the exponents of $\eta_{\tau_{p}}$ in the both sides of
(2.4), we see $J\supseteq M_{\tau_{p}}.$ For, if $J\not\supseteq
M_{\tau_{p}},$ then the exponent of $\eta_{\tau_{p}}$ in the right hand side
is smaller than $\sum_{i\in M_{\tau_{p}}}l(i,\tau_{p})$ which is the exponent
of $\eta_{\tau_{p}}$ in the left hand side, and this is a contradiction.

In summary, we have showed that

$J\cap(M_{\tau_{p}+1}\cup\cdots\cup M_{t})=\emptyset\quad\mbox{and}\quad
J\supseteq M_{\tau_{p}}.$

Cancelling the elements $\alpha_{i}\,(i\in M_{\tau_{p}})$ in the both sides of
(2.3), we get

$\prod_{i\in M_{\tau_{1}}\cup\cdots\cup M_{\tau_{p-1}}}\alpha_{i}=\prod_{i\in
J\setminus M_{\tau_{p}}}\alpha_{i}.$

As before, we substitute (2.1) into the above equation. By comparing the
exponents of $\eta_{\tau}\,(\tau_{p-1}<\tau<\tau_{p})$ in the both sides, we
see that

$J\cap\bigcup_{\tau_{p-1}<\tau<\tau_{p}}M_{\tau}=\emptyset.$

Then by comparing the exponents of $\eta_{\tau_{p-1}}$ in the both sides, we
see

$J\supseteq M_{\tau_{p-1}}.$

Then cancelling the elements $\alpha_{i}\,(i\in M_{\tau_{p}}\cup
M_{\tau_{p-1}})$ in the both sides of (2.3), we get

$\prod_{i\in M_{\tau_{1}}\cup\cdots\cup M_{\tau_{p-2}}}\alpha_{i}=\prod_{i\in
J\setminus(M_{\tau_{p}}\cup M_{\tau_{p-1}})}\alpha_{i}.$

Continuing this process, we finally get that

$J\cap\bigcup_{\tau_{i}<\tau<\tau_{i+1}}M_{\tau}=\emptyset\quad\mbox{and}\quad
J\supseteq M_{\tau_{i}},\quad 1\leq i\leq p,$

where we put $\tau_{p+1}:=t+1.$ Since $\\#J=\\#I,$ we see

$J=M_{\tau_{1}}\cup\cdots\cup M_{\tau_{p}}=I,$

which gives a contradiction.

$\bullet\quad\tau_{1}=0$ and $p\geq 2.$

The similar argument as before shows that

$J\cap\bigcup_{\tau_{i}<\tau<\tau_{i+1}}M_{\tau}=\emptyset\quad\mbox{and}\quad
J\supseteq M_{\tau_{i}},\quad 2\leq i\leq p.$

Then cancelling the elements $\alpha_{i}\,(i\in M_{\tau_{2}}\cup\cdots\cup
M_{\tau_{p}})$ in the both sides of (2.3), we get

$1=\prod_{i\in M_{0}}\alpha_{i}=\prod_{i\in
J\setminus(M_{\tau_{2}}\cup\cdots\cup M_{\tau_{p}})}\alpha_{i}.$

Then we see easily that

$J\cap\bigcup_{1\leq\tau<\tau_{2}}M_{\tau}=\emptyset.$

Because $\\#J=\\#I=m_{0}+m_{\tau_{2}}+\cdots+m_{\tau_{p}},$ we get

$J=M_{0}\cup M_{\tau_{2}}\cup\cdots\cup M_{\tau_{p}}=I,$

which gives a contradiction.

$\bullet\quad\tau_{1}=0$ and $p=1.$

In this case, (2.3) becomes

$1=\prod_{i\in J}\alpha_{i}.$

We see that $J\subseteq M_{0}.$ Then by $\\#J=\\#I,$ we get $J=M_{0}=I,$ which
also gives a contradiction.

In any case, we get a contradiction. Therefore the Claim 2.2 is proved. ∎

We shall prove next the following.

######  Claim 2.5.

$t=s-1$, $q=2s,$ and one of the following two cases occurs:

  1. (a)

$m_{0}=s+1$, $m_{1}=m_{2}=\dots=m_{s-1}=1;$

  2. (b)

$s$ is odd and $m_{0}=m_{1}=\dots=m_{s-1}=2.$

###### Proof.

Let $\sigma:\\{1,\dots,t\\}\to\\{1,\dots,t\\}$ be a bijection such that

$m_{\sigma(1)}\geq m_{\sigma(2)}\geq\cdots\geq m_{\sigma(t)}.$

We also put $\sigma(0)=0.$

Since $m_{\sigma(0)}+m_{\sigma(1)}+\cdots+m_{\sigma(t)}=q>s,$ there is a
minimum natural number $p$ such that

$m_{\sigma(0)}+m_{\sigma(1)}+\cdots+m_{\sigma(p)}>s.$

We consider the following two cases.

$\bullet\quad p=0.$

$p=0$ means that $m_{0}>s.$ Then since $t\geq s-1$ and every $m_{\tau}$
($1\leq\tau\leq t$) is a positive integer, we conclude that

$\displaystyle 2s\geq q$ $\displaystyle=m_{0}+m_{1}+\dots+m_{t}\geq(s+1)+t\geq
2s.$

It follows that $t=s-1$, $q=2s$ and $m_{0}=s+1$, $m_{1}=m_{2}=\dots=m_{t}=1.$
This shows the case (a) occurs. Hence Claim 2.5 holds when $p=0.$

$\bullet\quad p\geq 1.$

By the definition of $p$ and using Claim 2.2, we get

$m_{\sigma(0)}+m_{\sigma(1)}+\cdots+m_{\sigma(p)}\geq s+1$

and

$m_{\sigma(0)}+\cdots+m_{\sigma(p-1)}=:\tilde{m}\leq s-1.$

In particular, we see

$m_{\sigma(p)}\geq s+1-\tilde{m}\geq 2.$

Let $r:=\\#\big{\\{}\tau\in\\{1,\dots,t\\}\,|\,m_{\tau}=1\big{\\}}.$ Then
$0\leq r\leq t-p$ and

$m_{\sigma(t-r)}\geq 2,\quad m_{\sigma(t-r+1)}=\dots=m_{\sigma(t)}=1.$

By Claim 2.2, $\tilde{m}+r\leq s-1.$ For, if $\tilde{m}+r\geq s,$ by taking
$(s-\tilde{m})$ distinct indices $i_{1},\dots,i_{s-\tilde{m}}$ in
$\\{\sigma(t-r+1),\dots,\sigma(t)\\},$ we have

$\tilde{m}+m_{i_{1}}+\dots+m_{i_{s-\tilde{m}}}=s,$

which contradicts Claim 2.2. Thus

$r\leq s-1-\tilde{m}\leq m_{\sigma(p)}-2.$

Then by $t\geq s-1,$ we conclude that

$\displaystyle 2s\geq q$
$\displaystyle=m_{0}+m_{\sigma(1)}+\cdots+m_{\sigma(p)}+(m_{\sigma(p+1)}+\cdots+m_{\sigma(t-r)})+r$
$\displaystyle\geq m_{0}+p\cdot m_{\sigma(p)}+2(t-r-p)+r$
$\displaystyle=m_{0}+p\cdot m_{\sigma(p)}+2t-2p-r$ $\displaystyle\geq
m_{0}-2+(m_{\sigma(p)}-2)(p-1)+2s$ $\displaystyle\geq 2s.$

It follows that $t=s-1$, $q=2s$, $m_{0}=2$,
$m_{\sigma(1)}=\dots=m_{\sigma(p)}$, $r=s-1-\tilde{m}=m_{\sigma(p)}-2,$ and
$(m_{\sigma(p)}-2)(p-1)=0.$

If $m_{\sigma(p)}=2,$ then $r=0$ and $m_{\sigma(1)}=\dots=m_{\sigma(t)}=2.$ If
$m_{\sigma(p)}\neq 2,$ then $p=1$ and $m_{\sigma(1)}\geq 3.$ Thus
$r=m_{\sigma(1)}-2\geq 1$ and $m_{\sigma(1)}=s+1-\tilde{m}=s+1-m_{0}=s-1.$ It
follows that

$m_{\sigma(1)}+m_{\sigma(t)}=s,$

which contradicts Claim 2.2. So it is impossible that $m_{\sigma(p)}\neq 2,$
and thus we have $m_{0}=m_{1}=\dots=m_{t}=2.$

If $s$ is even, then

$m_{1}+m_{2}+\dots+m_{s/2}=s,$

which contradicts Claim 2.2. Thus $s$ is odd and the case (b) occurs. The
proof of Claim 2.5 is completed. ∎

By Claim 2.5, we know that $t=s-1$ and $q=2s.$ So, the conclusion (i) of Lemma
1.3 is true, and for proving the conclusion (ii) of Lemma 1.3, we need to show
that $(\alpha_{1},\dots,\alpha_{2s})$ is of the type (A) or type (B).

If the case (a) in Claim 2.5 occurs, then

$M_{1}=\\{u_{1}\\},\,M_{2}=\\{u_{2}\\},\dots,M_{s-1}=\\{u_{s-1}\\}\quad\mbox{and}\quad\\#M_{0}=s+1.$

We see that $\alpha_{i}=1$ for any
$i\in\\{1,\dots,2s\\}\setminus\\{u_{1},\dots,u_{s-1}\\}.$ Put

$\beta_{1}:=\alpha_{u_{1}},\,\beta_{2}:=\alpha_{u_{2}},\dots,\beta_{s-1}:=\alpha_{u_{s-1}}.$

Since

$\alpha_{u_{1}}=\eta_{1}^{l_{1}},\,\alpha_{u_{2}}=\eta_{2}^{l_{2}},\dots,\alpha_{u_{s-1}}=\eta_{s-1}^{l_{s-1}},$

where $\\{\eta_{1},\dots,\eta_{s-1}\\}$ is a basis of the $\mathbb{Q}$-vector
space
$\langle\alpha_{1},\dots,\alpha_{2s}\rangle\otimes_{\mathbb{Z}}\mathbb{Q},$ we
see that $\beta_{1},\dots,\beta_{s-1}$ are multiplicatively independent
elements in $G,$ and thus $\\{\beta_{1},\dots,\beta_{s-1}\\}$ is a basis of
$\langle\alpha_{1},\dots,\alpha_{2s}\rangle.$ After a suitable change of
indices, we have

$\alpha_{1}:\alpha_{2}:\dots:\alpha_{2s}=1:1:\dots:1:\beta_{1}:\dots:\beta_{s-1},$

which is a special case of the type (B).

Now assume the case (b) in Claim 2.5 occurs. We know that $s$ is odd and
$\\#M_{i}=2$ for any $0\leq i\leq s-1.$ We assume that

$M_{1}=\\{u_{1},v_{1}\\},\dots,M_{s-1}=\\{u_{s-1},v_{s-1}\\}\quad\mbox{and}\quad
M_{0}=\\{u_{0},v_{0}\\}.$

Regarding the $\alpha_{i}$ as elements in
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q},$ we
have

$\alpha_{u_{0}}=\alpha_{v_{0}}=1\quad\mbox{and}\quad\alpha_{u_{\tau}}=\eta_{\tau}^{l_{\tau}},\,\alpha_{v_{\tau}}=\eta_{1}^{l(v_{\tau},1)}\cdots\eta_{\tau}^{l(v_{\tau},{\tau})},\,1\leq\tau\leq
s-1,$ (2.6)

where $l(v_{\tau},{\tau})>0$ for any $1\leq\tau\leq s-1.$

We shall show that $\alpha_{u_{\tau}}=\alpha_{v_{\tau}}$ for any
$1\leq\tau\leq s-1.$

Noting that $(s-1)/2$ is a positive integer, we consider first the case when
$1\leq\tau\leq(s-1)/2.$ Put

$I:=\\{u_{\tau}\\}\cup M_{(s+1)/2}\cup\cdots\cup M_{s-1}.$

Then $\\#I=s,$ and by assumption, there exists a $J\subseteq\\{1,\dots,2s\\}$
with $\\#J=s$ and $J\neq I,$ such that

$\prod_{j\in I}\alpha_{j}=\prod_{j\in J}\alpha_{j}.$ (2.7)

Using (2.6), we represent the both sides of (2.7) with
$\eta_{1},\dots,\eta_{s-1}.$ For $i\in\\{(s+1)/2,\dots,s-1\\},$ by observing
the exponents of $\eta_{i}$ in the both sides of (2.7), we see $J\supseteq
M_{i}.$ Then by $\\#J=\\#I,$ we know that $J\cap(M_{0}\cup\cdots\cup
M_{(s-1)/2})$ has exactly one element, say $j_{0}.$ Cancelling the elements
$\alpha_{j}\,(j\in M_{(s+1)/2}\cup\cdots\cup M_{s-1})$ in the both sides of
(2.7), we get $\alpha_{u_{\tau}}=\alpha_{j_{0}}.$ Then by (2.6) and $J\neq I,$
we see $j_{0}=v_{\tau}.$ Therefore $\alpha_{u_{\tau}}=\alpha_{v_{\tau}}.$

Now let $(s+1)/2\leq\tau\leq s-1.$ Put

$I:=M_{0}\cup\cdots\cup M_{(s-3)/2}\cup\\{u_{\tau}\\}.$

Then $\\#I=s,$ and by assumption, there exists a $J\subseteq\\{1,\dots,2s\\}$
with $\\#J=s$ and $J\neq I,$ such that

$\prod_{j\in I}\alpha_{j}=\prod_{j\in J}\alpha_{j}.$ (2.8)

Using (2.6), we represent the both sides of (2.8) with
$\eta_{1},\dots,\eta_{s-1}.$ For
$i\in\\{(s-1)/2,\dots,s-1\\}\setminus\\{\tau\\},$ since the exponent of
$\eta_{i}$ in the left hand side of (2.8) is zero, we see $J\cap
M_{i}=\emptyset.$ Then by observing the exponents of $\eta_{\tau}$ in the both
sides of (2.8), we see $J\cap M_{\tau}$ has exactly one element. By
$\\#J=\\#I$ and $J\neq I,$ we get

$J=M_{0}\cup\cdots\cup M_{(s-3)/2}\cup\\{v_{\tau}\\}.$

Cancelling the elements $\alpha_{j}\,(j\in M_{0}\cup\cdots\cup M_{(s-3)/2})$
in the both sides of (2.8), we get $\alpha_{u_{\tau}}=\alpha_{v_{\tau}}.$

So we have proved that

$\alpha_{u_{\tau}}=\alpha_{v_{\tau}},\quad 1\leq\tau\leq s-1.$

Then as before, putting

$\beta_{\tau}:=\alpha_{u_{\tau}},\quad 1\leq\tau\leq s-1,$

we see $\\{\beta_{1},\dots,\beta_{s-1}\\}$ is a basis of
$\langle\alpha_{1},\dots,\alpha_{2s}\rangle,$ and after a suitable change of
indices, we have

$\alpha_{1}:\alpha_{2}:\dots:\alpha_{2s}=1:1:\beta_{1}:\beta_{1}:\beta_{2}:\beta_{2}:\cdots:\beta_{s-1}:\beta_{s-1},$

which shows $(\alpha_{1},\dots,\alpha_{2s})$ is of the type (A). This finishes
the proof for the Case $(\alpha).$

###  2.2 The proof for the Case $(\beta)$

Changing the indices of $\eta_{1},\dots,\eta_{t}$ if necessary, we may assume
that $l(i,t)<0$ for some $i\in\\{1,\dots,q\\}.$

Put

$\displaystyle N_{+}$
$\displaystyle:=\big{\\{}i\in\\{1,\dots,q\\}\,|\,l(i,t)>0\big{\\}},\quad
n_{+}:=\\#N_{+},$ $\displaystyle N_{0}$
$\displaystyle:=\big{\\{}i\in\\{1,\dots,q\\}\,|\,l(i,t)=0\big{\\}},\quad
n_{0}:=\\#N_{0},$ $\displaystyle N_{-}$
$\displaystyle:=\big{\\{}i\in\\{1,\dots,q\\}\,|\,l(i,t)<0\big{\\}},\quad
n_{-}:=\\#N_{-},$

where $l(i,t)$ are the integers in (2.1). Because $l(u_{t},t)=l_{t}>0,$ and
$M_{0}\subseteq N_{0},$ and $u_{1},\dots,u_{t-1}\in N_{0},$ we see that

$n_{+}\geq 1,\quad n_{-}\geq 1,\quad\mbox{and}\quad n_{0}\geq m_{0}+(t-1)\geq
t+1.$

Put $\tilde{n}:=\min\\{n_{+},n_{-}\\}$ and assume
$N_{0}=\\{p_{1},\dots,p_{n_{0}}\\}.$

######  Claim 2.9.

$2\leq s-\tilde{n}<n_{0}\leq 2(s-\tilde{n}),$ and the $n_{0}$-tuple
$(\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}})$ has the property
$(P_{n_{0},s-\tilde{n}}).$

###### Proof.

Since $q\leq 2s,$ we get

$2s\geq q=n_{0}+n_{+}+n_{-}\geq n_{0}+2\tilde{n},$

which implies $n_{0}\leq 2(s-\tilde{n}).$ By $t\geq s-1,$ we get

$n_{0}\geq t+1\geq s.$

Thus $n_{0}>s-\tilde{n}.$ Since $s\geq 3,$ we know that

$2s\geq n_{0}+n_{+}+n_{-}\geq 3+2\tilde{n},$

which implies $s-\tilde{n}\geq 2.$

Next we prove that $(\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}})$ has the
property $(P_{n_{0},s-\tilde{n}}).$

Consider first the case when $\tilde{n}=n_{+}.$ Take a subset $I$ of $N_{0}$
with $\\#I=s-n_{+}.$ Since $\\#(I\cup N_{+})=s,$ by assumption, there exists a
subset $J_{1}$ of $\\{1,\dots,q\\}$ with $\\#J_{1}=s$ and $J_{1}\neq(I\cup
N_{+}),$ such that

$\prod_{i\in(I\cup N_{+})}\alpha_{i}=\prod_{i\in J_{1}}\alpha_{i}.$ (2.10)

Regarding the $\alpha_{i}$ as elements in
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q}$ and
substituting (2.1) into (2.10), we represent the both sides of (2.10) with
$\eta_{1},\dots,\eta_{t}.$ Observing the exponents of $\eta_{t},$ we see

$J_{1}\supseteq N_{+}\quad\mbox{and}\quad J_{1}\cap N_{-}=\emptyset.$

Then $J_{1}\setminus N_{+}=:J\subseteq N_{0}$ and $\\#J=s-n_{+}.$ By
$J_{1}\neq(I\cup N_{+}),$ we also have $J\neq I.$ Cancelling the elements
$\alpha_{i}\,(i\in N_{+})$ in the both sides of (2.10), we get

$\prod_{i\in I}\alpha_{i}=\prod_{i\in J}\alpha_{i}.$

This shows $(\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}})$ has the property
$(P_{n_{0},s-n_{+}}).$

When $\tilde{n}=n_{-},$ similar argument shows that
$(\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}})$ has the property
$(P_{n_{0},s-n_{-}}).$ Therefore the Claim 2.9 is proved. ∎

######  Claim 2.11.

The subgroup $\langle\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\rangle$ of $G$
generated by $\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}$ is of rank $(t-1).$

###### Proof.

For any element
$\beta\in\langle\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\rangle,$ if we regard
$\beta$ as an element in
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q},$
then it can be represented by $\eta_{1},\dots,\eta_{t-1}.$ This fact implies
that any $t$ elements in
$\langle\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\rangle$ are multiplicatively
dependent, and therefore

${\rm rank}\\{\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\\}\leq t-1.$

On the other hand, we know
$\alpha_{u_{1}},\dots,\alpha_{u_{t-1}}\in\langle\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\rangle.$
And, as elements in
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q},$

$\alpha_{u_{1}}=\eta_{1}^{l_{1}},\dots,\alpha_{u_{t-1}}=\eta_{t-1}^{l_{t-1}}.$

Thus, $\alpha_{u_{1}},\dots,\alpha_{u_{t-1}}$ are multiplicatively independent
elements in $G$ and it follows that

${\rm rank}\\{\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\\}\geq t-1.$

So we get ${\rm rank}\\{\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\\}=t-1,$ which
proves Claim 2.11. ∎

By Claims 2.9 and 2.11, we can apply the induction hypothesis to the
$n_{0}$-tuple $(\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}})$ and conclude that

$t-1={\rm rank}\\{\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\\}\leq
s-\tilde{n}-1\leq s-2.$

Then by $t\geq s-1,$ we see

$t=s-1,\quad\tilde{n}=1,\quad\mbox{and}\quad{\rm
rank}\\{\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\\}=s-2.$

Applying the induction hypothesis to the $n_{0}$-tuple
$(\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}})$ again, we see that $n_{0}=2(s-1)$
and $(\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}})$ is of the type (A) or of the
type (B). So

$2s\geq q=n_{0}+n_{+}+n_{-}\geq 2(s-1)+2=2s.$

It follows that $q=2s$ and $n_{+}=n_{-}=1.$

We have proved that the conclusion (i) of Lemma 1.3 holds. We shall show next
that $(\alpha_{1},\dots,\alpha_{2s})$ is of the type (B), and this implies
that the conclusion (ii) of Lemma 1.3 also holds.

First, we show the following.

######  Claim 2.12.

It is impossible that the $(2s-2)$-tuple
$(\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}})$ is of the type (A).

###### Proof.

We assume the $(2s-2)$-tuple $(\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}})$ is of
the type (A). Then $(s-1)$ is odd and thus $s$ is an even integer with $s\geq
4.$

Since
$\\{1,\alpha_{u_{1}},\dots,\alpha_{u_{s-2}}\\}\subseteq\\{\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\\},$
when we regard the $\alpha_{i}$ as elements in
$\langle\alpha_{1},\dots,\alpha_{q}\rangle\otimes_{\mathbb{Z}}\mathbb{Q},$ we
may assume, after a suitable change of indices, that

$(\alpha_{1},\dots,\alpha_{2s})=(1,1,\eta_{1}^{l_{1}},\eta_{1}^{l_{1}},\dots,\eta_{s-2}^{l_{s-2}},\eta_{s-2}^{l_{s-2}},\eta_{s-1}^{l_{s-1}},\eta_{1}^{l^{\prime}_{1}}\cdots\eta_{s-1}^{l^{\prime}_{s-1}}),$
(2.13)

where $l_{1},\dots,l_{s-1}$ are positive integers in (2.1), and
$l^{\prime}_{1},\dots,l^{\prime}_{s-1}$ are integers and $l^{\prime}_{s-1}<0.$

For convenience’ sake, we write

$B_{0}:=\\{1,2\\},B_{1}:=\\{3,4\\},\dots,B_{s-2}:=\\{2s-3,2s-2\\},B_{s-1}:=\\{2s-1,2s\\}.$

We note that $s/2$ is a positive integer and $s/2\leq s-2.$ Put

$I:=B_{1}\cup\cdots\cup B_{s/2}.$

Since $\\#I=s,$ by assumption, there exists a $J\subseteq\\{1,\dots,2s\\}$
with $\\#J=s$ and $J\neq I,$ such that

$\prod_{j\in I}\alpha_{j}=\prod_{j\in J}\alpha_{j}.$

Substituting (2.13) into the above equation, we represent the both sides of
the above equation with $\eta_{1},\dots,\eta_{s-1}.$ If $J\cap
B_{s-1}=\emptyset,$ by observing the exponents of $\eta_{i}\,(1\leq i\leq
s/2),$ we see $J\supseteq B_{1}\cup\cdots\cup B_{s/2}$ and thus $J=I,$ which
is a contradiction. Therefore $J\cap B_{s-1}\neq\emptyset.$ Then by observing
the exponents of $\eta_{s-1},$ we see $J\supseteq B_{s-1}$ and
$l_{s-1}+l^{\prime}_{s-1}=0.$ By $\\#J=\\#I,$ there exists an index $i_{0}$
with $1\leq i_{0}\leq s/2$ such that $J\not\supseteq B_{i_{0}}.$ Then by
observing the exponents of $\eta_{i_{0}},$ it is necessary that
$l^{\prime}_{i_{0}}=l_{i_{0}}$ or $l^{\prime}_{i_{0}}=2l_{i_{0}}.$ In
particular, we get $l^{\prime}_{i_{0}}>0.$

Now we put

$I^{\prime}:=B_{0}\cup B_{1}\cup\cdots\cup B_{i_{0}-1}\cup
B_{i_{0}+1}\cup\cdots\cup B_{s/2}.$

By assumption, there exists a $J^{\prime}\subseteq\\{1,\dots,2s\\}$ with
$\\#J^{\prime}=s$ and $J^{\prime}\neq I^{\prime},$ such that

$\prod_{j\in I^{\prime}}\alpha_{j}=\prod_{j\in J^{\prime}}\alpha_{j}.$ (2.14)

Substituting (2.13) into (2.14), we represent the both sides of (2.14) with
$\eta_{1},\dots,\eta_{s-1}.$ If $J^{\prime}\cap B_{s-1}=\emptyset,$ by
observing the exponents of $\eta_{i}$ with $i\in\\{i_{0},s/2+1,\dots,s-2\\},$
we see

$J^{\prime}\cap(B_{i_{0}}\cup B_{s/2+1}\cup\cdots\cup B_{s-2})=\emptyset$

and thus $J^{\prime}=I^{\prime},$ which is a contradiction. Thus
$J^{\prime}\cap B_{s-1}\neq\emptyset,$ and as before we get
$J^{\prime}\supseteq B_{s-1}.$ Then the exponent of $\eta_{i_{0}}$ in the
right hand side of (2.14) is strictly greater than zero. However the exponent
of $\eta_{i_{0}}$ in the left hand side of (2.14) is zero. We get a
contradiction and thus the conclusion of Claim 2.12 holds. ∎

Now we know that the $(2s-2)$-tuple
$(\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}})$ is of the type (B).

Put

$\beta_{1}:=\alpha_{u_{1}},\,\beta_{2}:=\alpha_{u_{2}},\dots,\beta_{s-1}:=\alpha_{u_{s-1}}.$

Then $\beta_{1},\dots,\beta_{s-1}$ are multiplicatively independent elements
in $\langle\alpha_{1},\dots,\alpha_{2s}\rangle,$ and as elements in
$\langle\alpha_{1},\dots,\alpha_{2s}\rangle\otimes_{\mathbb{Z}}\mathbb{Q},$
they can be represented as follows:

$\beta_{1}=\eta_{1}^{l_{1}},\dots,\beta_{s-2}=\eta_{s-2}^{l_{s-2}},\,\beta_{s-1}=\eta_{s-1}^{l_{s-1}},$

where $l_{1},\dots,l_{s-1}$ are positive integers.

Since
$\\{\beta_{1},\dots,\beta_{s-2}\\}\subseteq\\{\alpha_{p_{1}},\dots,\alpha_{p_{n_{0}}}\\},$
by Observation 1.7 and suitably changing the indices of
$\eta_{1},\dots,\eta_{s-2}$ and of the $\alpha_{i}$’s, we may assume that

$\displaystyle(\alpha_{1},\dots,\alpha_{2s})$ $\displaystyle=\big{(}1,1,$
$\displaystyle\dots,\underset{\underset{\alpha_{s-k}}{\uparrow}}{1},\underset{\underset{\alpha_{s-k+1}}{\uparrow}}{\beta_{1}},\dots,\underset{\underset{\alpha_{2s-k-2}}{\uparrow}}{\beta_{s-2}},\underset{\underset{\alpha_{2s-k-1}}{\uparrow}}{(\beta_{1}\cdots\beta_{a_{1}})^{-1}},$
$\displaystyle\qquad(\beta_{a_{1}+1}\cdots\beta_{a_{2}})^{-1},\dots,\underset{\underset{\alpha_{2s-2}}{\uparrow}}{(\beta_{a_{k-1}+1}\cdots\beta_{a_{k}})^{-1}},\underset{\underset{\alpha_{2s-1}}{\uparrow}}{\eta_{s-1}^{l_{s-1}}},\underset{\underset{\alpha_{2s}}{\uparrow}}{\eta_{1}^{l^{\prime}_{1}}\cdots\eta_{s-1}^{l^{\prime}_{s-1}}}\big{)},$

where $0\leq k\leq s-2$, $1\leq a_{1}<\dots<a_{k}\leq s-2$, and
$l^{\prime}_{1},\dots,l^{\prime}_{s-1}$ are integers and $l^{\prime}_{s-1}<0.$

We shall prove next the following assertions which imply that
$(\alpha_{1},\dots,\alpha_{2s})$ is of the type (B):

$\begin{split}&l^{\prime}_{s-1}=-l_{s-1},\\\
&l^{\prime}_{i}\in\\{0,-l_{i}\\}\quad\mbox{for}\quad a_{k}+1\leq i\leq s-2,\\\
&l^{\prime}_{i}=0\quad\mbox{for}\quad 1\leq i\leq a_{k}.\end{split}$ (2.15)

Put

$I:=\\{1,2,\dots,s-k,\,2s-k-1,\dots,2s-2\\}.$

Since $\\#I=s,$ by assumption, there exists a $J\subseteq\\{1,\dots,2s\\}$
with $\\#J=s$ and $J\neq I,$ such that

$\prod_{j\in I}\alpha_{j}=\prod_{j\in J}\alpha_{j}.$

We represent the both sides of the above equation with
$\eta_{1},\dots,\eta_{s-1}.$ If $J\cap\\{2s-1,2s\\}=\emptyset,$ by observing
the exponents of $\eta_{i}\,(1\leq i\leq s-2),$ we see
$J\cap\\{s-k+1,\dots,2s-k-2\\}=\emptyset,$ and thus $J=I,$ which is a
contradiction. Therefore $J\cap\\{2s-1,2s\\}\neq\emptyset.$ Then by observing
the exponents of $\eta_{s-1},$ we see $J\supseteq\\{2s-1,2s\\}$ and
$l_{s-1}+l^{\prime}_{s-1}=0.$ Assume $l^{\prime}_{i}\neq 0$ for some
$a_{k}+1\leq i\leq s-2.$ Since the exponent of $\eta_{i}$ in the left hand
side is zero, we see that $s-k+i\in J$ and $l_{i}+l^{\prime}_{i}=0.$ Assume
that $l^{\prime}_{i}>0$ for some $1\leq i\leq a_{k},$ then we see that the
exponent of $\eta_{i}$ in the right hand side is strictly greater than that in
the left hand side and this is a contradiction.

In summary, we have proved that $l^{\prime}_{s-1}=-l_{s-1},$ and
$l^{\prime}_{i}\in\\{0,-l_{i}\\}$ for $a_{k}+1\leq i\leq s-2,$ and
$l^{\prime}_{i}\leq 0$ for $1\leq i\leq a_{k}.$

We shall show next that, for any $1\leq i\leq a_{k},$ it is impossible that
$l^{\prime}_{i}<0.$ Without loss of generality, we assume that
$l^{\prime}_{1}<0.$ Put

$I^{\prime}:=\\{1,2,\dots,s-k,\,s-k+1,\,2s-k,\dots,2s-2\\}.$

By assumption, there exists a $J^{\prime}\subseteq\\{1,\dots,2s\\}$ with
$\\#J^{\prime}=s$ and $J^{\prime}\neq I^{\prime},$ such that

$\prod_{j\in I^{\prime}}\alpha_{j}=\prod_{j\in J^{\prime}}\alpha_{j}.$

We represent the both sides of the above equation with
$\eta_{1},\dots,\eta_{s-1}.$ Assume first that
$J^{\prime}\cap\\{2s-1,2s\\}=\emptyset.$ By observing the exponents of
$\eta_{i}$ with $i\in\\{1,a_{1}+1,a_{1}+2,\dots,s-2\\},$ we see

$J^{\prime}\cap\\{s-k+a_{1}+1,\,s-k+a_{1}+2,\dots,2s-k-1\\}=\emptyset.$

Then since $2s-k-1\not\in J^{\prime},$ by observing the exponents of
$\eta_{i}\,(2\leq i\leq a_{1}),$ we see
$J^{\prime}\cap\\{s-k+2,\dots,s-k+a_{1}\\}=\emptyset.$ Thus by
$\\#J^{\prime}=\\#I^{\prime},$ we get $J^{\prime}=I^{\prime},$ which is a
contradiction. Therefore $J^{\prime}\cap\\{2s-1,2s\\}\neq\emptyset,$ and as
before, we get $J^{\prime}\supseteq\\{2s-1,2s\\}.$ Since $l^{\prime}_{1}<0,$
we see that the exponent of $\eta_{1}$ in the right hand side is strictly less
than that in the left hand side. We get a contradiction and thus (2.15) is
proved.

By (2.15), we conclude that there is a subset $\Lambda$ of
$\\{a_{k}+1,\dots,s-1\\}$ with $s-1\in\Lambda$ such that

$\alpha_{2s}=(\prod_{i\in\Lambda}\beta_{i})^{-1}.$

Thus, suitably changing indices, we have

$\displaystyle\alpha_{1}:\alpha_{2}:\dots:\alpha_{2s}$
$\displaystyle=1:1:\dots:1:\beta_{1}:\dots:\beta_{s-1}:$
$\displaystyle(\beta_{1}\cdots\beta_{a_{1}})^{-1}:\dots:(\beta_{a_{k-1}+1}\cdots\beta_{a_{k}})^{-1}:(\beta_{a_{k}+1}\cdots\beta_{a_{k+1}})^{-1},$

where $1\leq k+1\leq s-1$, $1\leq a_{1}<\dots<a_{k}<a_{k+1}\leq s-1,$ and
$\\{\beta_{1},\dots,\beta_{s-1}\\}$ is a basis of
$\langle\alpha_{1},\dots,\alpha_{2s}\rangle.$ This shows that
$(\alpha_{1},\dots,\alpha_{2s})$ is of the type (B) and therefore we complete
the proof of Lemma 1.3.

## References

  * [1] Hirotaka Fujimoto.  The uniqueness problem of meromorphic maps into the complex projective space.  Nagoya Math. J., 58:1–23, 1975.
  * [2] Hirotaka Fujimoto.  A uniqueness theorem of algebraically non-degenerate meromorphic maps into $P^{N}(\mathbf{C})$.  Nagoya Math. J., 64:117–147, 1976.
  * [3] Hirotaka Fujimoto.  Remarks to the uniqueness problem of meromorphic maps into $P^{N}(\mathbf{C})$, I.  Nagoya Math. J., 71:13–24, 1978.
  * [4] Rolf Nevanlinna.  Einige Eindeutigkeitssätze in der Theorie der meromorphen Funktionen.  Acta Math., 48(3-4):367–391, 1926.
  * [5] George Pólya.  Bestimmung einer ganzen Funktion endlichen Geschlechts durch viererlei Stellen.  Mat. Tidsskrift B, pages 16–21, 1921.  https://zbmath.org/?q=an%3A48.0354.03.


# Subradiance and Superradiant Long Range Excitation Transport among Quantum
Emitter Ensembles in a Waveguide

Martin Fasser  Institut für Theoretische Physik, Universität Innsbruck,
Technikerstr. 21a, A-6020 Innsbruck, Austria  Laurin Ostermann  Institut für
Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020
Innsbruck, Austria  Helmut Ritsch  Institut für Theoretische Physik,
Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria  Christoph
Hotter<EMAIL_ADDRESS>Institut für Theoretische Physik,
Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria

(August 28, 2024)

###### Abstract

In contrast to free space, in waveguides the dispersive and dissipative
dipole-dipole interactions among quantum emitters exhibit a periodic behavior
over remarkably long distances. We propose a novel setup exploiting this long-
range periodicity in order to create highly excited subradiant states and
facilitate fast controlled collective energy transport amongst far-apart
ensembles coupled to a waveguide. For sufficiently large ensembles collective
superradiant emission into the fiber modes dominates over its free space
counterpart. We show that for a large number of emitters a fast transverse
coherent pulse can create almost perfect subradiant states with up to $50\%$
excitation. On the other hand, for a coherent excitation of one sub-ensemble
above an overall excitation fraction of $50\%$ we find a nearly lossless and
fast energy transfer to the ground state sub-ensemble. This transport can be
enhanced or suppressed by controlling the positions of the ensembles relative
to each other, while it can also be realized with a random position
distribution. In the optimally enhanced case this fast transfer appears as
superradiant emission with subsequent superabsorption, yet, without a
superradiant decay after the absorption. The highly excited subradiant states
as well as the superradiant excitation transfer appear as suitable building
blocks in applications like active atomic clocks, quantum batteries, quantum
information protocols and quantum metrology procedures such as fiber-based
Ramsey schemes.

##  I Introduction

Spatial confinement of electromagnetic radiation field modes increases the
atom-field coupling as well as the resulting dipole-dipole interaction between
distant quantum emitters as the coupling is proportional to the field
amplitude per photon generated by one emitter at the position of another.
Extreme cases are optical resonators exhibiting quasi-uniform strong coupling
over the entire mode volume Dovzhenko  _et al._ (2018) or single mode
waveguides González-Tudela  _et al._ (2024) such as optical fibers with weaker
but close to infinite range interaction at their center.

As the atoms in a fiber are often distributed over large distances, their
direct local interaction via free space is typically very small compared to
the collective interaction mediated by the fiber Ostermann  _et al._ (2019).
Recent advances in fiber fabrication technology, laser cooling and
manipulation of cold atomic ensembles allow for the implementation of such
collective dynamics at unprecedented atom numbers Ruks  _et al._ (2024); Chen
_et al._ (2024); Ribezzo  _et al._ (2023). While one typically first observes
superradiant collective emission Dicke (1954); Gross and Haroche (1982) into
the fiber Goban  _et al._ (2015); Okaba  _et al._ (2019); Pennetta  _et al._
(2022); Liedl  _et al._ (2024), in this work we find that the opposite case of
subradiance Albrecht  _et al._ (2019); Holzinger  _et al._ (2022); Zanner  _et
al._ (2022); Tiranov  _et al._ (2023) as well as superabsorption Higgins  _et
al._ (2014); Yang  _et al._ (2021); Quach  _et al._ (2022) can be implemented
and reveals a plethora of novel physical phenomena and applications,
especially in very large ensembles.

In addition to constituting a novel implementation of protected Ramsey
spectroscopy Ostermann  _et al._ (2013, 2014); Hotter  _et al._ (2023); Bohr
_et al._ (2024) we predict fast efficient long-distance collective energy
transfer Hama  _et al._ (2018a, b) based on superradiance with subsequent
superabsorption. This interplay between subradiant states with strongly
reduced emission rates and the superradiant energy transfer can be exploited
in quantum memories Zhao  _et al._ (2008a, b); Lvovsky  _et al._ (2009);
Plankensteiner  _et al._ (2015); Asenjo-Garcia  _et al._ (2017), quantum
batteries Ferraro  _et al._ (2018); Pirmoradian and Mølmer (2019); Quach and
Munro (2020); Ferioli  _et al._ (2021); Quach  _et al._ (2022) and in order to
remotely prepare perfectly inverted ensembles Dias  _et al._ (2021).

Figure 1: _Model._ Two ensembles of atoms coupled to a waveguide mode.
Initially, only one ensemble with $N_{\mathrm{p}}$ atoms is pumped while the
other with $N_{\mathrm{np}}$ atoms remains in the ground state. Below a
certain pump threshold an almost perfectly subradiant state is created, while
above this threshold a fast superradiant energy transfer from the pumped to
the non-pumped ensemble takes place, where the energy is (super-)absorbed and
remains in the second ensemble.

The manuscript has the following structure: after introducing our general
model based on quantum emitter arrays with fiber-mediated effective dipole-
dipole interactions we focus on the special case of two far-apart ensembles.
In order to treat a large number of emitters and high excitation manifolds we
employ cumulant expansion equations up to second order to numerically study
generic cases of the collective system dynamics, when only one of two
ensembles is coherently pumped. We then analyze the transition from a
subradiant collective decay blockade to a superradiant energy transfer between
the two ensembles as a function of atom numbers, coupling conditions and
initial state.

##  II Model

We consider $N$ identical two-level emitters coupled to a single optical fiber
mode, see fig. 1. The coupling rate of each emitter into the fiber is given by
$\Gamma^{1\mathrm{D}}=\Gamma\frac{k_{0}\mu^{2}}{\epsilon_{0}A}\chi$, which is
determined by the free space decay rate $\Gamma$, the effective area $A$, the
mode wavenumber $k_{0}$ and the normalization constant $\chi$ that accounts
for the distribution of the electric field into the mode Ostermann  _et al._
(2019); Cardenas-Lopez  _et al._ (2023). In the rotating frame of the atomic
transition frequency, the system is described by the Hamiltonian

$H=\sum_{i\neq j}^{N}\Omega_{ij}^{1\mathrm{D}}\sigma^{+}_{i}\sigma^{-}_{j}.$
(1)

The waveguide-mediated dipole-dipole coupling rates are

$\Omega^{1\mathrm{D}}_{ij}(x_{ij})=\frac{\Gamma^{1\mathrm{D}}}{2}\sin{(k_{\mathrm{eff}}x_{ij})},$
(2)

where $x_{ij}=|x_{i}-x_{j}|$ is the distance between two emitters $i$ and $j$,
and $k_{\mathrm{eff}}$ denotes the effective wavenumber of the guided mode.
The incoherent part describing the dissipative processes is accounted for by
the Lindblad term

$\mathcal{L}[\rho]=\frac{1}{2}\sum_{i,j}^{N}\Gamma^{1\mathrm{D}}_{ij}(2\sigma^{-}_{i}\rho\sigma^{+}_{j}-\sigma^{+}_{i}\sigma^{-}_{j}\rho-\rho\sigma^{+}_{i}\sigma^{-}_{j})$
(3)

with the collective fiber-mediated decay rates

$\Gamma^{1\mathrm{D}}_{ij}(x_{ij})=\Gamma^{1\mathrm{D}}\cos{(k_{\mathrm{eff}}x_{ij})}.$
(4)

We assume sufficiently large distances between the emitters such that we can
neglect free space dipole-dipole interaction, which decreases as $1/x$ in the
far field. Furthermore, we are only interested in time scales, where the
individual free space spontaneous emission of the atoms is not relevant. The
open system can thus be described by the following master equation in the
Born-Markov approximation Gardiner and Zoller (2004)

$\dot{\rho}=i[\rho,H]+\mathcal{L}[\rho].$ (5)

Since we aim at considering large systems with up to $10^{6}$ emitters, where
a full quantum simulation is by far out of reach, we will treat this problem
in a second-order cumulant expansion Kubo (1962); Plankensteiner  _et al._
(2022) which turns out to capture the dynamics remarkably well. A code example
that automatically derives and numerically solves the corresponding equations
is provided in the appendix.

##  III Subradiant States

Figure 2: _Subradiant states._ (a) Percentage of lost excitation for a full
excitation of the number of pumped atoms $N_{\mathrm{p}}$. For large $N$ the
fraction of lost excitation approaches zero. (b) Absolute number of lost
excitations for different fractions of pumped atoms $N_{\mathrm{p}}/N$. Below
the threshold, the number of lost excitations saturates to
$N_{\mathrm{p}}/(N_{\mathrm{np}}-N_{\mathrm{p}})$. (d) Lost excitation
fraction for coherent excitations to an initial excited state population of
$\langle\sigma_{\mathrm{p}}^{ee}\rangle(t_{0})$ for $N=10^{4}$ atoms. Weaker
coherent excitations can lead to more losses. (e) Lost excitation for an
initial $2/3\pi$-pulse. It does not saturate for large $N$. (c) and (f)
Population of the Dicke states for $N=20$ atoms and an initial $\pi$-pulse and
$2/3\pi$-pulse, respectively on $N_{\mathrm{p}}=8$ atoms. In (c) the
subradiant states (shaded red) and the collective decay (merely reducing $M$)
are indicated.

Subradiant states with respect to a single mode waveguide are of interest for
various quantum technologies, in particular for controlled photon storage and
release Asenjo-Garcia  _et al._ (2017); Holzinger  _et al._ (2022). However,
deterministically and reliably creating such states is experimentally
difficult. In ref. Holzinger  _et al._ (2022) the authors showed that quasi-
localized dark states can be created by exciting a minority fraction of the
emitters separated by multiples of half an effective wavelength
$\lambda_{\mathrm{eff}}/2$. In this case, the coherent coupling between the
emitters vanishes ($\Omega_{ij}^{\mathrm{1D}}=0$), and the dynamics reduces to
waveguide-mediated collective emission. The mere presence of the coupled
emitters in the ground state inhibits the superradiant decay of the excited
fraction. So far only systems with a moderate number of emitters where a small
fraction is fully excited have been considered. With a larger fraction of
excited emitters the prepared states lose a certain amount of their excitation
energy before the system becomes fully dark.

In the following, we analyze the corresponding dynamics in the limit of large
numbers of emitters and investigate the scaling of the lost excitation
fraction in the preparation process of these subradiant states via transverse
coherent pumping. We start by assuming a spacing of $\lambda_{\mathrm{eff}}$
between the emitters. The discrete translation symmetry of the model allows us
to describe it with two collective spins only, a spin-$N_{\mathrm{p}}/2$ for
the pumped atoms and a spin-$N_{\mathrm{np}}/2$ for the non-pumped atoms. In
fig. 2a we show the percentage of lost excitation depending on the total
number of emitters $N=N_{\mathrm{p}}+N_{\mathrm{np}}$ coupled to the waveguide
and the fraction of fully excited atoms $N_{\mathrm{p}}/N$. We observe that
this percentage decreases with $N$. In particular, the suppression with
growing $N$ dominates over the increased losses for larger fractional
excitations. This means, that the percentage of lost excitations can be made
arbitrarily small for a sufficiently large number of emitters, up to the
threshold for subradiance at $N_{\mathrm{p}}/N=50\%$.

The number of lost excitations and the subradiance threshold can be nicely
understood by means of the Dicke states $|J,M\rangle$ Dicke (1954), the
eigenstates of the collective spin operators $J^{z}|J,M\rangle=M|J,M\rangle$
and $\mathbf{J}^{2}|J,M\rangle=J(J+1)|J,M\rangle$, with $0\leq J\leq N/2$ and
$|M|\leq J$. Figure 2c depicts the Dicke state representation for a full
excitation of $N_{\mathrm{p}}=8$ atoms and
$N=N_{\mathrm{np}}+N_{\mathrm{p}}=20$. We obtain a distribution of Dicke
states with a well-defined $M$ quantum number corresponding to
$N_{\mathrm{p}}$ excitations. Since the collective decay of a Dicke state
appears as a reduction of $M$ with $J$ being unchanged
($J^{-}|J,M\rangle\rightarrow|J,M-1\rangle$), one finds that all states with
$M=-J$ are fully subradiant. It is also clear that only subradiant states with
$M\leq 0$ can exist, corresponding to $N_{\mathrm{p}}\leq N/2$ Shammah  _et
al._ (2018); Shankar  _et al._ (2021); Holzinger  _et al._ (2022); Hotter  _et
al._ (2023); Bohr  _et al._ (2024). The average lost excitations then
correspond to the population of the state $|\psi\rangle$ times its excitation
difference to the subradiant state with the same $J$ quantum number to which
it will decay, i.e. $\sum_{J,M}|\langle\psi|J,M\rangle|^{2}(M+J)$. In fig. 2b
we show the total number of lost excitations for different fractions of pumped
atoms $N_{\mathrm{p}}/N$. We find that it converges to
$N_{\mathrm{p}}/(N_{\mathrm{np}}-N_{\mathrm{p}})$ for large $N$. The dotted
lines correspond to calculations using the Dicke states, which perfectly agree
with full quantum simulations (not shown). The solid lines are the results of
the second-order cumulant expansions, which agree very well with the full
quantum model, especially for large $N$.

In fig. 2f we show the population of the Dicke states for an initial coherent
excitation of $N_{\mathrm{p}}=8$ atoms with a $2/3\pi$-pulse
($\langle\sigma_{\mathrm{p}}^{ee}\rangle(t_{0})=75\%$) and $N=20$.
Surprisingly, this state loses more excitations than with a $\pi$-pulse,
although it is initially less excited. This is shown in Figure 2d and Figure
3c for different initial coherent excitations and the number of pumped
emitters. Figure 2e depicts the lost excitation as a function of the atom
number $N$ for an initial $2/3\pi$-pulse. In this case, it does not saturate
for large $N$. This means, for a spacing between the emitters in multiples of
$d=\lambda_{\mathrm{eff}}$ this dark state preparation protocol works best
when a fraction of the emitters are fully excited. For a coherent excitation,
this requires an almost perfect $\pi$-pulse.

Another way of creating subradiant states is to use opposite phases such that
the emitted radiation interferes destructively Ballantine and Ruostekoski
(2020); Rubies-Bigorda and Yelin (2022); Hotter  _et al._ (2023); Yan  _et
al._ (2023); Bohr  _et al._ (2024). In a waveguide, this can be realized by a
spacing of the atoms in multiples of $d=\lambda_{\mathrm{eff}}/2$. In fig. 3f
we plot the percentage of lost excitation for this case. One can see that the
lost excitation depends mainly on the initial excitation
$\mathcal{E}=\langle\sigma^{ee}_{\mathrm{p}}\rangle(t_{0})\cdot
N_{\mathrm{p}}/N$. Below the threshold at $\mathcal{E}=1/2$ the lost
excitation fraction is very close to zero for a sufficiently large $N$, i.e.
we can create subradiant states through an arbitrary coherent excitation
almost without losing energy (for $\mathcal{E}<1/2$).

Figure 3: _Steady-State Population and Lost Excitation._ Scan over the
fraction of pumped atoms $N_{\mathrm{p}}/N$ and the initial excited state
population $\langle\sigma^{ee}_{\mathrm{p}}\rangle(t_{0})$ after a coherent
excitation pulse. The upper panel depicts the case for an emitter spacing of
$d=\lambda_{\mathrm{eff}}$ ($\Gamma^{\mathrm{1D}}_{ij}=\Gamma^{\mathrm{1D}}$,
symmetric decay rates) and the lower the one for $d=\lambda_{\mathrm{eff}}/2$
($\Gamma^{\mathrm{1D}}_{ij}=\pm\Gamma^{\mathrm{1D}}$, anti-symmetric decay
rates). (a) and (d) show the steady-state population of the pumped atoms, (b)
and (e) of the non-pumped atoms while (c) and (f) depict the fraction of lost
excitations. The subradiant behavior and the energy transfer drastically
improve for the anti-symmetric decay rates. For an initial full excitation,
the behavior is the same for both cases. For the lower panel a threshold at
$\mathcal{E}=1/2$ (dashed white line) from subradiance to superradiance with
subsequent superabsorption can be observed prominently. The chosen atom number
in these scans is $N=10^{6}$.

##  IV Superradiance and Superabsortion

Figure 4: _Time Evolution._ (a) Subradiant behavior (solid lines) and
superradiant energy transfer (dashed lines) for initially fully inverted
pumped atoms. (b) Coherent excitation to $75\%$ ($2/3\pi$-pulse) for symmetric
decay rates ($d=\lambda_{\mathrm{eff}}$), where subradiance and energy
transfer are less optimal. (c) Anti-symmetric decay rates
($d=\lambda_{\mathrm{eff}}/2$), featuring subradiance and the optimal energy
transfer for arbitrary coherent excitations. The chosen atom number is
$N=10^{4}$, for $\mathcal{E}<1/2$ ($\mathcal{E}>1/2$) we use
$N_{\mathrm{p}}=2\cdot 10^{3}$ ($N_{\mathrm{p}}=8\cdot 10^{3}$).

In the previous section, we have focused on the collective subradiant behavior
of the emitters coupled to a waveguide. From the Dicke triangle
representation, it is clear that such subradiant states can only exist for a
total number of excitations below $N/2$ (corresponding to $M\leq 0$). In the
following, we investigate the dynamics of total initial excitations above
$50\%$. As before, we start by considering a spacing of multiples of
$\lambda_{\mathrm{eff}}$. In this case, we observe a fast excitation transfer
from the initially fully excited to the non-excited ensemble facilitated by
the collective decay via the shared bath, i.e. the waveguide Hama  _et al._
(2018b, a); Dias  _et al._ (2023). In fig. 4a, we show a typical time
evolution above the threshold ($\mathcal{E}>1/2$, dashed lines), where we
observe a superradiant decay of the initially excited ensemble and
superabsorption in the initially non-excited ensemble. For comparison we have
also plotted the behavior below the threshold ($\mathcal{E}<1/2$, solid
lines), where no energy transfer takes place.

Superabsorption can be interpreted as the inverse process to superradiance
Higgins  _et al._ (2014); Yang  _et al._ (2021); Quach  _et al._ (2022).
Hence, the excitation is usually immediately reemitted after the rapid
collective absorption, which makes it difficult to observe the phenomenon. In
the two-ensemble case described above, however, the emitters are in a
subradiant state after the absorption, which means it is not followed by
superradiant emission. This should make it experimentally much more accessible
and useful. Let us mention that in the effective waveguide description the
term superabsorption refers to the fast absorption of virtual photons.

Similar to the subradiance described in the previous section we find that the
superradiance with subsequent superabsorption in the case of a spacing of
multiples of $\lambda_{\mathrm{eff}}$ works well only if one ensemble is
initially fully excited. Figure 4b shows the dynamics for a coherent
excitation of the pumped ensemble to
$\langle\sigma_{\mathrm{p}}^{\mathrm{ee}}\rangle(t_{0})=75\%$
($2/3\pi$-pulse), where we notice that the energy transfer is not fully
exciting the non-pumped ensemble above threshold ($\mathcal{E}>1/2$, dashed
lines). This can also be seen in the Scan 3b.

Again, in order to optimize this process such that it works well for arbitrary
coherent excitations, we can use an emitter spacing of multiples of
$\lambda_{\mathrm{eff}}/2$. The resulting opposite phases in the collective
decay lead to a sharp threshold at an initial excitation fraction of
$\mathcal{E}=1/2$. Below the threshold the system is subradiant and above we
observe optimal superradiance from the pumped ensemble with subsequent
superabsorption from the non-pumped ensemble. This is demonstrated in the time
evolution in fig. 4c and scans in fig. 3d-f, the dashed line represents the
threshold at $50\%$ total excitation.

Comparing the results for $\lambda_{\mathrm{eff}}$ and
$\lambda_{\mathrm{eff}}/2$ spacing in fig. 3a-c and fig. 3d-f, respectively,
we observe a profound difference in the threshold behavior for not completely
inverted coherently excited ensembles.

In order to characterize the superradiant energy transfer fig. 5a depicts the
atom number dependence of the time $T_{\mathrm{sa}}$ it takes the non-pumped
ensemble to reach $95\%$ of the final state excitation, i.e.
$\langle\sigma^{ee}_{\mathrm{np}}\rangle=0.95\cdot\langle\sigma^{ee}_{\mathrm{np}}\rangle(t_{\mathrm{end}})$.
We find a scaling close to $T_{\mathrm{sa}}\sim N^{-1}$ as for superradiant
decay Dicke (1954); Norcia  _et al._ (2016); Laske  _et al._ (2019); Pennetta
_et al._ (2022). For an initial excitation sufficiently above the threshold
this scaling holds for any coherent excitation. Close to the threshold
different coherent excitations with the same initial excitation fraction
$\mathcal{E}$ can lead to a slightly different transfer duration
$T_{\mathrm{sa}}$. This can be seen in fig. 5b, where we plot
$T_{\mathrm{sa}}(\mathcal{E})$ for different coherent excitations. In the
vicinity of the threshold, the transfer is faster for a full excitation of
fewer emitters (data points with smaller $T_{\mathrm{sa}}$) than a smaller
coherent excitation of more emitters with the same $\mathcal{E}$.

Figure 5: _Superabsorption Time Scale._ (a) Atom number scaling of the energy
transfer time $T_{\mathrm{sa}}$ as approximately $N^{-1}$ (b) Transfer time
$T_{\mathrm{sa}}$ for different coherent excitations with the same initial
excitation fraction
$\mathcal{E}=\langle\sigma_{\mathrm{p}}^{ee}\rangle(t_{0})N_{\mathrm{p}}/N$.
Far above the threshold $T_{\mathrm{sa}}$ depends on $\mathcal{E}$ only. Close
to the threshold, a full excitation of fewer emitters leads to a faster
transfer than a smaller coherent excitation of more emitters. The number of
atoms in (b) is $N=10^{4}$.

##  V Uniform Distribution: Coherent Dipole-Dipole Interaction Effects

Figure 6: _Time Evolution including Coherent Dipole-Dipole Interaction._
Population of the pumped and non-pumped emitters. For an initial full
excitation of the pumped ensemble (a), the dynamics are only slightly modified
by the coherent interaction. In contrast the behavior for a coherent
excitation to $75\%$ changes significantly due to the coherent interaction
(b). (c) shows that the optimal behavior can be recovered with a vanishing
excitation phase in the pumped ensemble for coherent excitations. The atom
numbers are the same as in Figure 4.

So far we have considered the special cases in which the emitters are located
precisely at multiples of $\lambda_{\mathrm{eff}}/2$, where the coherent
dipole-dipole interaction vanishes, i.e. $\Omega_{ij}=0$ for all $i$ and $j$.
For a moderate number of emitters that can be positioned e.g. with optical
tweezers or in the case of microwave waveguides this is experimentally
feasible. However, for large atomic clouds that couple to an optical
waveguide, all positions along the waveguide with respect to the periodic
coupling are equally likely. In order to investigate the influence of the
resulting coherent dipole-dipole interaction we sample the different
positions. In particular, we use four equidistant positions with a spacing of
$d=\lambda_{\mathrm{eff}}/4$ for the pumped and non-pumped ensemble. A
comparison with six equidistant positions is shown in fig. 7a, b, d and e in
the appendix, which underpins that four ensembles are sufficient for a
qualitative description of the coherent dipole-dipole interaction effects.

In fig. 6 we show the dynamics for $N=10^{4}$ emitters for different cases.
For a fully excited pumped ensemble (Figure 6a) the behavior below as well as
above threshold is only weakly modified by the coherent dipole-dipole
interaction. The slower overall dynamic mainly stems from the reduced
incoherent rates. This has been tested by simulating the dynamics with
incoherent rates $\Gamma_{ij}$ corresponding to a spacing
$d=\lambda_{\mathrm{eff}}/4$ while neglecting the coherent interaction
($\Omega_{ij}=0$), see fig. 7c and f in the appendix. In contrast, for an
arbitrary coherent excitation, the subradiant and superabsorption features are
strongly affected by the coherent dipole-dipole interaction. The dynamics for
an initial coherent $2/3\pi$-pulse is shown in fig. 6b. We see that the
behavior is not optimal, similar to the one with a spacing of
$d=\lambda_{\mathrm{eff}}$ (Figure 4). The coherent interaction seems to
cancel the positive effect from the opposite phases of the collective decay
rates.

The crucial difference in the above two cases is the initial coherence in the
later one. This means an initial incoherent excitation to
$\langle\sigma_{\mathrm{p}}^{ee}\rangle(t_{0})=75\%$ should lead to the same
optimal subradiant and superradiant transfer behavior. One way to
approximately resemble an incoherent excitation is to excite ensembles with
vanishing phases. In fig. 6c we show the dynamics for this case, where the
pumped ensemble is divided into two sub-ensembles which are excited with
opposite phases. This leads to an almost optimal behavior below and above the
threshold. Note that for a large ensemble of emitters an overall vanishing
excitation phase is usually automatically given due to the position
distribution. These results suggest that the phenomenon should be observable
with two separated dilute clouds of atoms coupled to a waveguide.

##  VI Conclusions

We have shown a transition from sub- to superradiance with subsequent
superabsorption for two ensembles of emitters coupled to a waveguide. The lost
excitation for the subradiant states saturates for a large number of emitters.
This allows for creating highly excited subradiant states with up to $50\%$
excitations. Above the threshold an efficient superradiant energy transfer
between the two ensembles takes place. We have demonstrated that for a not
fully inverting coherent excitation of the pumped atoms, a vanishing phase is
crucial for optimal behavior. This is, however, usually automatically given
for large atomic ensembles. The necessary operating conditions are within
reach of current experimental setups Okaba  _et al._ (2019); Abend  _et al._
(2023); Bertoldi  _et al._ (2021); Melli  _et al._ (2024).

The efficiency of the process can be increased by optimizing the emitter
positions, this can e.g. be realized with optical tweezers Pesce  _et al._
(2020) or self-organization Chang  _et al._ (2013); Ritsch  _et al._ (2013).
The described mechanisms should also be valid for multi-mode waveguides. In
future work, it could be interesting to investigate potential retardation
effects for widely separated ensembles, as well as the possibility of
engineering the dissipation by detuning ensembles. We also want to mention
that ensembles of emitters in a single-mode bad cavity, where the cavity field
can be adiabatically eliminated, feature a similar collective dissipation with
opposite phases. Yet, for a vanishing atom-cavity detuning the resulting
effective equations for the atoms do not experience coherent dipole-dipole
interaction terms Gardiner and Zoller (2004); Shankar  _et al._ (2021); Jäger
_et al._ (2022). This can be beneficial for the above-described phenomena. Of
particular interest are ring cavities with a uniform coupling strength for all
atoms Zhang  _et al._ (2023).

## Acknowledgments

The numerical simulations were performed with the open-source frameworks
QuantumCumulants.jl Plankensteiner  _et al._ (2022) and QuantumOptics.jl
Krämer  _et al._ (2018). M. F. and H. R. acknowledge funding from the Austrian
Science Fund (FWF) doctoral college DK-ALM W1259-N27. M. F., L. O. and H. R.
acknowledge the FET OPEN Network Cryst3 funded by the European Union (EU) via
Horizon 2020.

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## Appendix

###  VI.1 Simulation Code

The Julia code to derive and numerically solve the second-order cumulant
equations can be found in the online documentation of the toolbox
QuantumCumulants.jl Plankensteiner  _et al._ (2022):
https://qojulia.github.io/QuantumCumulants.jl/stable/examples/waveguide/. By
setting $M_{\mathrm{p}}=4$ and $M_{\mathrm{np}}=4$ in the example one creates
the 324 equations to reproduce the results in Figure 6a and b.

###  VI.2 Six vs. Four Subensembles

In fig. 7 we compare the time evolution including coherent dipole-dipole
interaction for four and six different positions at
$x_{4}=\\{0,1,2,3\\}\cdot\lambda_{\mathrm{eff}}/4$ and
$x_{6}=\\{0,1,2,3,4,5\\}\cdot\lambda_{\mathrm{eff}}/6$, respectively. For the
full excitation in fig. 7a and b the difference is barely visible. For an
initial coherent excitation to $75\%$ the oscillations are stronger, see
Figure 7d and e. However, we expect the oscillations to damp out very quickly
for more positions. Unfortunately, we cannot test this in the second-order
cumulant expansion approach, since the number of coupled differential
equations for the $2\times 6$ spins is already $702$. In fig. 7c we show the
dynamics for each of the four ensembles. In fig. 7f we set the coherent
dipole-dipole interaction to zero, and see that this only slightly changes the
dynamics.

Figure 7: _Coherent Dipole-Dipole Interaction Effects._ (a) and (d) Time
evolution above and below the threshold for an initial $\pi$-pulse and
$2/3\pi$-pulse excitation, respectively. These plots are the same as in Figure
6 for four positions with a spacing of $d=\lambda_{\mathrm{eff}}/4$. (b) and
(e) Comparison to six sub-ensembles with a spacing of
$d=\lambda_{\mathrm{eff}}$/6. The time evolution in (a) and (b) is almost
identical, for (d) and (e) they agree qualitatively. In (c) we plot all
trajectories of the different ensembles for the case presented in (a). In (f)
we artificially set the coherent dipole-dipole interaction to zero. This
reveals that the slower dynamic for the four positions can be traced back to
the reduced collective decay rate.