# The boundary of bordified Outer space

Karen Vogtmann

###### Abstract.

We study the boundary of the “Jewel space" $\mathcal{J}_{n}$ constructed in
[4]. This is an equivariant deformation retract of Outer space $CV_{n}$ on
which $Out(F_{n})$ acts properly and cocompactly, and is homeomorphic to the
Bestvina-Feighn bordification of $CV_{n}$. In the current paper we analyze the
structure of the boundary of $\mathcal{J}_{n}$. We then use the desctiption of
the simplicial closure $CV_{n}^{*}$ as the sphere complex of a connected sum
of $n$ copies of $S^{1}\times S^{2}$ to prove that this boundary is homotopy
equivalent to the subcomplex of $CV_{n}^{*}$ spanned by vertices at infinity.

## Introduction

The group $Out(F_{n})$ acts properly but not cocompactly on the space $CV_{n}$
known as Outer space in rank $n$. In [2] Bestvina and Feighn defined a partial
compactification $\widehat{CV}_{n}$ of $CV_{n},$ analogous to the Borel-Serre
bordification of symmetric spaces, on which $Out(F_{n})$ acts both properly
and cocompactly. They proved that their bordification is $(2n-5)$-connected at
infinity, which allowed them to conclude that $Out(F_{n})$ is a
$(2n-3)$-dimensional virtual duality group. This means that there is an
$Out(F_{n})$-module $D$ and isomorphisms

$H^{i}(\Gamma,M)\cong H_{2n-3-i}(\Gamma,D\otimes M)$

for any torsion-free subgroup $\Gamma$ of finite index in $Out(F_{n})$ and any
$\Gamma$-module $M$. The module $D$ is called the dualizing module for
${\mathrm{Out}}(F_{n})$, and is isomorphic to
$H^{2n-3}({\mathrm{Out}}(F_{n}),{\mathbb{Z}}({\mathrm{Out}}(F_{n})))$ (see
[3], Proposition 11.3).

In [4] Bux, Smillie and the author constructed a deformation retract
$\mathcal{J}_{n}$ of $CV_{n}$ on which $Out(F_{n})$ acts properly and
cocompactly and showed that $\mathcal{J}_{n}$ is $(2n-5)$-connected at
infinity, giving an alternate proof that $Out(F_{n})$ is a virtual duality
group. They also proved that $\mathcal{J}_{n}$ is in fact homeomorphic to
Bestvina and Feighn’s bordification $\widehat{CV}_{n}$.

The Borel-Serre bordification $\widehat{X}$ of the symmetric space $X$
associated to a connected linear algebraic group $G$ defined over
${\mathbb{Q}}$ is a manifold with corners. If $\Gamma$ is an arithmetic
subgroup of $G$ then the dualizing module for $\Gamma$ can be identified with
the cohomology with compact supports of $\widehat{X}.$ By Poincaré-Lefschetz
duality, the cohomology with compact supports of $\widehat{X}$ is isomorphic
to the relative homology of $\widehat{X}$ modulo its boundary, which in turn
is isomorphic to the homology of $\partial\widehat{X}$ (shifted by 1) since
$\widehat{X}$ is contractible. The boundary $\partial\widehat{X}$ has a
covering by Euclidean spaces, whose nerve is the Tits building for
$G({\mathbb{Q}})$. Finally, by the Solomon-Tits theorem, this Tits building is
homotopy equivalent to a bouquet of spheres of dimension equal to the virtual
cohomological dimension of $\Gamma$.

A closely analogous picture was established by J. Harer for the mapping class
group $Mod(\Sigma)$ of a closed orientable surface $\Sigma$ [10]. Here the
symmetric space is replaced by the Teichmüller space of $\Sigma$ and the Tits
building by the curve complex $\mathcal{C}(\Sigma)$, so that the homology of
the curve complex (which, like the Tits building, is homotopy equivalent to a
bouquet of spheres) is a dualizing module for $Mod(\Sigma)$.

For $Out(F_{n})$ the space $CV_{n}$ serves as an analog of the symmetric
space; in particular the cohomology with compact supports of
$\widehat{CV}_{n}$ is the dualizing module for $Out(F_{n})$. However
$\widehat{CV}_{n}$ is not a manifold so Poincaré-Lefschetz duality is not
available, and the cohomology with compact supports of $\widehat{CV}_{n}$
cannot be identified with the cohomology of its boundary. Nevertheless in this
note we will show that parts of the analogy are still valid. Specifically, we
study the Jewel space model $\mathcal{J}_{n}$ for the bordification and show
that its boundary is covered by products of Jewel spaces for smaller graphs;
in particular these products are contractible, with contractible
intersections. We then show that the nerve of this cover can be identified
with the subcomplex $CV_{n}^{\infty}$ spanned by vertices at infinity of the
simplicial closure $CV_{n}^{*}$ of $CV_{n}$. At the end of the paper we ask
whether the analogy goes further, in particular whether $CV_{n}^{\infty}$ is
homotopy equivalent to a bouquet of spheres.

##  1\. The jewel space $\mathcal{J}_{n}$

Recall that a core graph is a graph with no isolated vertices or separating
edges. It may have loops or mutiple edges, and it may be disconnected or have
bivalent vertices. By a subgraph of $G$ we mean the subgraph spanned by a non-
empty set of edges (so it cannot have isolated vertices), and a core subgraph
is a subgraph which is itself a core graph. Let $\sigma(G)$ be the regular
Euclidean simplex whose vertices correspond to edges of $G$, so that subgraphs
$H$ correspond to faces $\sigma(H)$ of $\sigma(G)$. If $G$ is a core graph, we
defined the jewel $J(G)$ to be the convex polytope obtained from $\sigma(G)$
by shaving faces opposite core subgraphs by appropriate constants (see [4],
Section 2). An example is shown in Figure 1.

$1$$2$$4$$3$$e_{2}$$e_{3}$$e_{4}$$e_{1}$ Figure 1. Example of a graph and its
associated jewel

###  1.1. Facets of a jewel

In this section we examine the codimension one faces (the facets) of a jewel
$J(G)$.

######  Notation.

Let $H$ be a subgraph of $G$.

  * •

$G/\mkern-6.0mu/H$ is the graph obtained from $G$ by collapsing each edge of
$H$ to a point.

  * •

$H^{\vee}$ is the subgraph of $G$ spanned by the edges in $G-H$.

If $e$ is a single edge of $G$ which is not a loop then $J(G)$ has a facet
(opposite the vertex corresponding to $e$) isomorphic to
$J(G/\mkern-6.0mu/e)$. These facets are contained in the boundary of
$\sigma(G)$ and are called interior facets. All other facets correspond to
core graphs $C$ of $G,$ and are called border facets. We will denote the
border facet opposite $\sigma(C)$ by $f_{C}$. The following lemma records the
fact that the border facets of $J(G)$ have a recursive structure.

######  Lemma 1.1.

The border facet $f_{C}$ of $J(G)$ is isomorphic to $J(C)\times
J(G/\mkern-6.0mu/C)$.

Note that both $C$ and $G/\mkern-6.0mu/C$ are core graphs, so $J(C)$ and
$J(G/\mkern-6.0mu/C)$ are defined.

###### Proof.

By Proposition 7.1 of [4], the codimension $(t+k)$ faces of $J(G)$ are
uniquely determined by sets $S=\\{e_{1},\ldots,e_{t},C_{1},\ldots,C_{k}\\}$,
where

  1. (1)

the $e_{i}$ are edges of $G$ and $F=e_{1}\cup\ldots\cup e_{t}$ is a forest in
$G$

  2. (2)

$C_{1}\subset\ldots\subset C_{k}$ is a chain of proper core subgraphs, and

  3. (3)

$C_{i}$ is the core of $F\cup C_{i}$.

The face $f_{S}$ determined by $S$ is a face of $f_{T}$ if and only if
$S\supset T$, so that $J(G)$ has an alternate description as the geometric
realization of the poset of such sets, ordered by inclusion.

Let $C$ be a core graph and $S=\\{C\\}$ the corresponding codimension one
border facet, i.e. the face $f_{C}$ opposite $\sigma(C)$. The isomorphism
$f_{C}\cong J(C)\times J(G/\mkern-6.0mu/C)$ is given by the poset map

$\\{F,C_{0}\subset\ldots\subset C_{i-1}\subset C\subset
C_{i+1}\subset\ldots\subset C_{k}\\}\mapsto\hbox{\hskip 113.81102pt}$
$\hbox{\hskip 85.35826pt}(\\{F\cap C,C_{0}\subset\ldots\subset
C_{i-1}\\},\\{F/\mkern-6.0mu/(F\cap
C),C_{i+1}/\mkern-6.0mu/C\subset\ldots\subset C_{k}/\mkern-6.0mu/C\\}).$

The first factor can clearly be identified with $J(C)$. To see that the second
can be identified with $J(G/\mkern-6.0mu/C)$ observe that

  1. (1)

the image of $F$ in $G/\mkern-6.0mu/C$ is still a forest since $C=core(F\cup
C)$,

  2. (2)

the image $C_{i}/\mkern-6.0mu/C$ is still a core graph, which is

  3. (3)

equal to the core of $F/\mkern-6.0mu/(F\cap C)\cup C_{i}/\mkern-6.0mu/C=(F\cup
C_{i})/\mkern-6.0mu/C$.

∎

The facet $f_{C}$ is close to, and parallel to, the face $\sigma(C^{\vee}),$
so it is sometimes more natural to label it by $H=C^{\vee}$; in this case we
will denote it by $f^{H}$. Thus$f^{H}$ is only defined when $H^{\vee}$ is a
core graph $C$, in which case $f^{H}=f_{C},$ and we say that this facet is
opposite $\sigma(C)$ and faces $\sigma(H)$.

##  2\. Walls in the boundary of $\mathcal{J}_{n}$

Jewel space $\mathcal{J}_{n}$ is the union of all marked jewels $J(g,G),$ and
is homeomorophic to the Bestvina-Feighn bordification $\widehat{CV}_{n}$ by
[4]. The border facets form the boundary of $\mathcal{J}_{n}$. In this section
we cover this boundary by contractible subcomplexes with contractible
intersections. This is analogous to Borel and Serre’s covering of the
bordification of symmetric space by Euclidean spaces $e(P)$ associated to
parabolic subgroups $P$. The nerve of Borel and Serre’s covering is homotopy
equivalent to the associated spherical Tits building. In our case the nerve of
the covering is homotopy equivalent to a subcomplex of the simplicial closure
of Outer space, which we will identify with a certain sphere complex. We begin
by briefly reviewing the relevant definitions.

###  2.1. The sphere complex and Outer space

Outer space $CV_{n}$ decomposes as a union of open simplices $\sigma(G,g)$,
one for each equivalence class of marked graphs $(G,g)$. However, the faces of
some of these simplices are not in $CV_{n}$, namely the faces obtained by
collapsing a subgraph that contains a loop, so this does not give $CV_{n}$ the
structure of a simplicial complex. However, adding all of the missing faces
does give a simplicial complex, known as the simplicial closure $CV_{n}^{*}$
of Outer space. A good way to understand $CV_{n}^{*}$ is by identifying open
simplices in $CV_{n}$ with open simplices in the sphere complex of the
3-manifold $M_{n}=\\#_{n}(S^{1}\times S^{2}).$ The simplicial closure
$CV_{n}^{*}$ can then be defined as the sphere complex itself. We refer the
reader to [11] for details, but sketch the ideas below.

An embedded $2$-sphere in a $3$-manifold $M$ is trivial if it bounds a ball or
is parallel to a boundary component. A sphere system is a set of isotopy
classes of non-trivial $2$-spheres which has a set of pairwise-disjoint
representatives. The sphere complex $\mathscr{S}(M)$ is the simplicial complex
with a $k$-simplex $\sigma(\mathcal{S})$ for each sphere system $\mathcal{S}$
in $M$ with $k+1$ elements.

######  Definition 2.1.

A sphere system $\mathcal{S}$ is complete if all components of the manifold
$M-\mathcal{S}$ obtained by cutting $M$ open along $\mathcal{S}$ are simply-
connected.

In the case $M=M_{n}$ this means that each component is homeomorphic to a
punctured ball. Note that a complete system must have at least $n$ spheres,
and a complete system with exactly $n$ spheres has exactly one complementary
component. Every maximal sphere system cuts $M_{n}$ into 3-punctured balls, so
in particular is complete.

######  Proposition 2.2 ([11], Appendix).

Outer space $CV_{n}$ is the union of the interiors of simplices
$\sigma(\mathcal{S})$ in $\mathscr{S}(M_{n})$ for $\mathcal{S}$ complete.

The translation from the original description of Outer space as a space of
marked metric graphs $(G,g)$ goes as follows. Using barycentric coordinates on
the simplices $\sigma(\mathcal{S})$, a point in $\mathscr{S}(M_{n})$ can be
thought of as a weighted complete sphere system, where the weight on each
sphere is a positive real number, and the sum of the weights is equal to 1.
Dual to any sphere system in $M_{n}$ is a graph embedded in $M_{n}$, with one
vertex in each component of $M_{n}-\mathcal{S}$ and one edge piercing each
sphere in $\mathcal{S}$; for example, the dual graph to a minimal complete
system is a rose. If we fix an identification of $\pi_{1}(M_{n})$ with
$F_{n}$, and if all components of $M-\mathcal{S}$ are simply-connected, the
embedding of this dual graph into $M_{n}$ identifies its fundamental group
with $F_{n}$ up to conjugacy, i.e. it determines a marking of the dual graph.
The weight on the sphere intersecting a given edge can be interpreted as the
length of the edge; thus a sphere system determines a marked metric graph with
volume one.

In the other direction, given a marked metric graph, put a point in the middle
of each edge, fatten the graph to a handlebody (so the points become disks),
double the handlebody to get a manifold homeomorphic to $M_{n},$ (so the disks
become $2$-spheres). Now use the marking to construct a homeomorphism to
$M_{n}$. The images of the $2$-spheres are a sphere system in $M_{n}$.

One advantage of this description of $CV_{n}$ is that a single sphere can be a
member of many different, incompatible sphere systems, i.e. an edge of a
marked graph has a life independent of any particular marked graph containing
it.

Another advantage is that the proof that $CV_{n}$ is contractible using
surgery paths in sphere systems [13] works with only minor modifications for
the analogous space $CV_{n,s}$ defined using
$M_{n,s}=M_{n}\backslash\coprod_{i=1}^{s}B_{s}$, the manifold obtained by
removing $s$ disjoint balls from $M_{n}.$ Then $CV_{n,s}$ is equal to the
subspace of $\mathscr{S}(M_{n,s})$ consisting of the union of the interiors of
simplices $\sigma(\mathcal{S})$ for $\mathcal{S}$ complete. This can also be
interpreted as a space of marked graphs with leaves; to see the idea, take the
dual graph of a sphere system, then attach a leaf for each boundary sphere
$\mathfrak{s}$, emanating from the vertex corresponding to the component of
$M_{n,s}-\mathcal{S}$ that contains $\mathfrak{s}$.

Any diffeomorphism of a 3-manifold $M$ sends disjoint $2$-spheres to disjoint
$2$-spheres, so acts on $\mathscr{S}(M)$. By a theorem of Laudenbach [14],
$Out(F_{n})$ is isomorphic to $\pi_{0}(\hbox{Diff}(M_{n}))/D$, where $D$ is a
finite group which acts trivially on isotopy classes of $2$-spheres ($D$ is
generated by “Dehn twists in $2$-spheres"). Under the identification of the
space of marked graphs with the space of sphere systems, the induced action of
$Out(F_{n})$ on $\mathscr{S}(M_{n})$ becomes the old action of $Out(F_{n})$ on
marked graphs. For $s>0$, define $\Gamma_{n,s}$ to be the group of isotopy
classes of diffeomorphisms of $M_{n,s}$ fixing the boundary, modulo Dehn
twists on $2$-spheres (or, equivalently, the group of homotopy classes of
homotopy equivalences of a rose with $s$ leaves that fix the leaves). Then
$\Gamma_{n,s}$ acts on $\mathscr{S}(M_{n,s})$, and the stabilizer of a point
permutes the spheres (preserving weights), so is always finite. In addition to
the fact that $\Gamma_{n,0}\cong Out(F_{n})$, we also have $\Gamma_{n,1}\cong
Aut(F_{n})$ .

######  Definition 2.3.

The space $CV_{n,s}$ is the union of the interiors of simplices
$\sigma(\mathcal{S})$ in $\mathscr{S}(M_{n,s})$ for $\mathcal{S}$ complete.

Note that $CV_{n,s}$ is not a subcomplex of $\mathscr{S}(M_{n,s})$, just a
subspace, since each simplex is missing some faces (in particular, all of its
vertices are missing). The entire simplicial complex $\mathscr{S}(M_{n,s})$ is
called the simplicial closure of $CV_{n,s}$ and we will also use the notation
$CV_{n,s}^{*}$.

###  2.2. Core systems

We next reinterpret the notions of core graph and core subgraph in terms of
sphere systems. We are thinking of a point in $CV_{n,s}$ as a point in the
interior of some $\sigma(\mathcal{S})$, for $\mathcal{S}$ a complete sphere
system in $M_{n,s}$.

######  Definition 2.4.

A sphere system $\mathcal{S}=\\{\mathfrak{s}_{0},\ldots,\mathfrak{s}_{k}\\}$
in $M_{n,s}$ is a core system if each $\mathfrak{s}_{i}$ is non-separating in
$M_{n,s}$.

Let $\mathcal{O}_{n,s}$ be the union of the interiors of the simplices
$\sigma(\mathcal{S})$ for $S$ complete and core. The group $\Gamma_{n,s}$ acts
properly on $\mathcal{O}_{n,s}$.

######  Proposition 2.5.

For $n\geq 1$, $CV_{n,s}$ equivariantly deformation retracts to
$\mathcal{O}_{n,s}$.

###### Proof.

If $n\geq 1$ a complete system must contain at least one non-separating
sphere. Removing a separating sphere from a complete sphere system results in
another complete sphere system, by Van Kampen’s theorem. Thus the map which
linearly decreases the weights on all separating spheres to zero, compensating
by linearly increasing the weights on all non-separating spheres, gives an
equivariant deformation retraction from all of $CV_{n,s}$ to
$\mathcal{O}_{n,s}$. ∎

######  Remark 2.6.

Both [2] and [4] use $\mathcal{O}_{n}=\mathcal{O}_{n,0}$, which is sometimes
called reduced Outer space, instead of $CV_{n}$

We now describe the jewel retract $\mathcal{J}_{n,s}$ in the language of
sphere systems. Since the jewel associated to $J(G,g)$ is obtained by
truncating certain faces of $\sigma(G,g)$ by amounts satisfying certain
constraints, we begin by identifying the relevant faces and constants.

######  Definition 2.7.

Let $\mathcal{S}$ be a core sphere system in $M=M_{n,s}$. A subsystem
$\mathcal{T}\subset\mathcal{S}$ is a core complement if
$\mathcal{S}-\mathcal{T}$ is core in $M-\mathcal{T}$.

For any sphere system $\mathcal{S}$ in $M=M_{n,s}$, define $h(\mathcal{S})$ to
be the number of spheres in $\mathcal{S}$ minus the number of complementary
components:

$h(\mathcal{S})=\\#\mathcal{S}-\hbox{rank}(H_{0}(M-\mathcal{S})).$

(In terms of graphs, $h(\mathcal{S})$ is “edges minus vertices," i.e. the
negative of the Euler characteristic of the graph dual to $\mathcal{S}$.)

######  Example 2.8.

If $\mathcal{S}=\\{\mathfrak{s}\\}$ is a single non-separating sphere, then
$h(\mathfrak{s})=0$. If $\mathcal{S}$ is complete, then $h(\mathcal{S})=n-1$.

If $\mathcal{T}$ is a core complement in $\mathcal{S}$, let
$r(\mathcal{S},\mathcal{T})=h(\mathcal{S})-h(\mathcal{T})\hbox{ and }$

$t(\mathcal{S},\mathcal{T})=\frac{3^{r(\mathcal{S},\mathcal{T})}}{N}$

for some fixed $N\gg n$.

We can now construct the jewel of a (not necessarily complete) core sphere
system:

######  Definition 2.9.

Let $\mathcal{S}$ be a core sphere system. The jewel
$J(\mathcal{S})\subset\sigma(\mathcal{S})$ is the polytope obtained by
truncating the face $\sigma(\mathcal{T})$ whenever
$\mathcal{T}\subset\mathcal{S}$ a core complement, by the amount
$t(\mathcal{S},\mathcal{T})$. The codimension one face of $J(\mathcal{S})$
obtained by truncating $\sigma(\mathcal{T})$ is called the facet
$\varphi(\mathcal{S},\mathcal{T})$.

######  Lemma 2.10.

Let $G$ be the graph dual to $\mathcal{S}$. Then $J(G,g)$ is isometric to
$J(\mathcal{S})$.

###### Proof.

The subsystem $\mathcal{T}\subset\mathcal{S}$ is a core complement if and only
if the edges dual to $\mathcal{S}-\mathcal{T}$ form a core graph in the graph
dual to $\mathcal{S}$. Furthermore, the truncation constants satisfy the
conditions used in the proof of Proposition 2.2 of [4], which characterizes
the jewel $J(G,g)$. ∎

If we glue together all jewels associated to complete systems in $M_{n,s}$ we
obtain the jewel space $\mathcal{J}_{n,s}=\mathcal{J}(M_{n,s}),$ as explained
in [4]. We are interested in studying the boundary
$\partial\mathcal{J}_{n,s},$ i.e. the union of the facets
$\varphi(\mathcal{S},\mathcal{T})$ for all complete core $\mathcal{S}$ and
core complements $\mathcal{T}\subset\mathcal{S}$.

######  Lemma 2.11.

Let $\mathcal{T}$ be an incomplete core system in $M$. Then $\mathcal{T}$ is a
core complement for a core system $\mathcal{S}$ if and only if
$\mathcal{S}-\mathcal{T}$ is a core system in $M-\mathcal{T}$.

###### Proof.

This follows from the definition of core complement, together with the
observation that if a sphere in $\mathcal{S}-\mathcal{T}$ does not separate
$M-\mathcal{T}$, then it does not separate $M$. ∎

If $\mathcal{T}$ is any subsystem of a core system $\mathcal{S}$, then
$\mathcal{S}$ is complete in $M$ if and only if $\mathcal{S}-\mathcal{T}$ is
complete in $M-\mathcal{T}$. The lemma thus shows that there are lots of
complete core systems $\mathcal{S}$ containing a given incomplete system
$\mathcal{T}$ as a core complement.

The boundary $\partial\mathcal{J}_{n,s}$ is the union of walls
$W(\mathcal{T})$ for $\mathcal{T}$ incomplete, where

$W(\mathcal{T})=\bigcup_{\mathcal{S}\hbox{ \tiny complete core, }\\\
\mathcal{S}\supset\mathcal{T}}\varphi(\mathcal{S},\mathcal{T})$

To understand $W(\mathcal{T})$, we translate Lemma 1.1 into the language of
sphere systems:

######  Lemma 2.12.

Let $\mathcal{T}$ be a core complement to a core system $\mathcal{S}$ in $M$.
Then

$\varphi(\mathcal{S},\mathcal{T})\cong J_{M}(\mathcal{T})\times
J_{M-\mathcal{T}}(\mathcal{S}-\mathcal{T}).\qed$

######  Proposition 2.13.

For any incomplete core system $\mathcal{T}$ in $M$, $W(\mathcal{T})$ is
contractible.

###### Proof.

By Lemma 2.12 we have

$W(\mathcal{T})\cong J_{M}(\mathcal{T})\times\bigcup_{{}_{\mathcal{S}\hbox{
\tiny complete core, }\\\
\mathcal{S}\supset\mathcal{T}}}J_{M-\mathcal{T}}(\mathcal{S}-\mathcal{T})\cong
J_{M}(\mathcal{T})\times\mathcal{J}({M-\mathcal{T}}).$

$J_{M}(\mathcal{T})$ is a truncated simplex, so is contractible. If
$M-\mathcal{T}$ has $k$ components $M^{1}\cup\ldots\cup M^{k}$, then
$\mathcal{J}(M-\mathcal{T})\cong\mathcal{J}(M^{1})\ast\ldots\ast\mathcal{J}(M^{k})$.
Since each $\mathcal{J}(M^{i})$ is a deformation retract of some
$CV_{n_{i},s_{i}}$, each is contractible. ∎

######  Proposition 2.14.

$\partial\mathcal{J}_{n,s}$ is homotopy equivalent to the subcomplex
$\mathscr{S}_{\infty}(M_{n,s})$ of the complex $\mathscr{S}(M_{n,s})$of core
sphere systems spanned by incomplete systems.

###### Proof.

Note that any core complement $\mathcal{T}$ is necessarily incomplete, since
any sphere in $M-\mathcal{T}$ for a complete system $\mathcal{T}$ is
separating. Therefore letting $N\to\infty$ in the definition of the truncation
constant $t(\mathcal{S},\mathcal{T}),$ so that $t(\mathcal{S},\mathcal{T})\to
0,$ gives a map

$\partial\mathcal{J}_{n,s}\to\mathscr{S}_{\infty}(M_{n,s}).$

By Proposition 2.13 the pre-image of an open simplex $\sigma(\mathcal{T})$ in
$\mathscr{S}_{\infty}(M_{n,s})$ is homeomorphic to
$(J_{M}(\mathcal{T}))^{0}\times\mathcal{J}({M-\mathcal{T}})$, where
$(J_{M}(\mathcal{T}))^{0}$ denotes the interior of the jewel
$J_{M}(\mathcal{T})$. Since these pre-images are all contractible, the map is
a homotopy equivalence. ∎

######  Remark 2.15.

The subcomplex $\mathscr{S}_{\infty}(M_{n})$ contains the entire
$(n-2)$-skeleton of $\mathscr{S}(M_{n}),$ since it takes at least $n$ spheres
to cut $M_{n}$ into simply connected pieces. Since
$\mathscr{S}_{\infty}(M_{n})$ is contractible, this shows that
$\mathscr{S}_{\infty}(M_{n})$ is at least $(n-3)$-connected.

It is easy to compute, using the Euler characteristic, that
$\mathscr{S}_{\infty}(M_{n})$ is $(3n-6)$-dimensional. This situation reflects
that of the arc complex $\mathcal{A}$ of a surface with boundary. The complex
$\mathcal{A}$ embeds naturally into $\mathscr{S}(M_{n})$ for appropriate $n$
(take the product of the surface with an interval to get a handlebody, then
double it; the arcs become discs, then spheres). For example, if the surface
has genus $g>0$ with one boundary component, the entire arc complex
$\mathcal{A}$ is contractible of dimension $6g-4$, the subcomplex
$\mathcal{A}_{\infty}$ at infinity is $(6g-6)$-dimensional and
$\mathcal{A}_{\infty}$ contains the $(2g-2)$-skeleton so is
$(2g-3)$-connected. Harer proves that $\mathcal{A}_{\infty}$ is in fact
homotopy equivalent to a bouquet of $(2g-2)$-spheres [10], and we ask whether
the same is true here:

Question: Is $\mathscr{S}_{\infty}(M_{n})$ homotopy equivalent to a bouquet of
spheres of dimension $n-2$?

Recall that ${\mathrm{Out}}(F_{n})$ is a virtual duality group with virtual
cohomological dimension $2n-3$, so the dualizing module is
$H^{2n-3}(Out(F_{n});{\mathbb{Z}}(Out(F_{n}))\cong
H^{2n-3}_{c}(\mathcal{J}_{n})$. The space $\mathcal{J}_{n}$ has dimension
$3n-4$, so if $\mathcal{J}_{n}$ were an orientable manifold Poincaré duality
would identify $H^{2n-3}_{c}(\mathcal{J}_{n})$ with
$H_{n-1}(\mathcal{J}_{n},\partial\mathcal{J}_{n})$, which is isomorphic to
$H_{n-2}(\partial\mathcal{J}_{n})\cong H_{n-2}(\mathscr{S}_{\infty}(M_{n}))$.
But $\mathcal{J}_{n}$ is not a manifold, and we ask

Question: What (if any) is the relation between
$H_{n-2}(\mathscr{S}_{\infty}(M_{n}))$ and $H^{2n-3}_{c}(\mathcal{J}_{n})$?

Both of these questions can also be asked for $\mathscr{S}_{\infty}(M_{n,s})$
(where the spheres would have dimension $n-3+s$) and $\mathcal{J}_{n,s}$.

###  2.3. The Jacobian map

There is a natural map, sometimes called the Jacobian  or Torelli map, from
$CV_{n}$ to the symmetric space $Q_{n}={\mathrm{SL}}(n,\mathbb{R})/SO(n)$ of
positive definite quadratic forms on ${\mathbb{R}}^{n}$. It is easy to
describe using the description of a point of $CV_{n}$ as a graph $X$ together
with a marking $g$ which identifies $\pi_{1}(X)$ with $F_{n}$. If we equip the
Euclidean space $\mathbb{R}^{E}$ spanned by the edges $E$ of $X$ with the
standard inner product, we can restrict this to the subspace
$H_{1}(X)\subset{\mathbb{R}}^{E}$ to get a positive definite quadratic form on
$H_{1}(X)$. The marking $g$ identifies $H_{1}(X)$ with $\mathbb{R}^{n}$, so
this form is a point of $Q_{n}$.

Question: Does the Jacobian map send the Bestvina-Feighn bordification to the
Borel-Serre bordification? What is the image of $\mathcal{J}_{n}$? What is the
image of $\partial\mathcal{J}_{n}$?

## References

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# Environment-mediated entropic uncertainty in charging quantum batteries

Meng-Long Song  School of Physics & Optoelectronic Engineering, Anhui
University, Hefei 230601, People’s Republic of China  Li-Juan Li  School of
Physics & Optoelectronic Engineering, Anhui University, Hefei 230601, People’s
Republic of China  Xue-Ke Song<EMAIL_ADDRESS>School of Physics &
Optoelectronic Engineering, Anhui University, Hefei 230601, People’s Republic
of China  Liu Ye  School of Physics & Optoelectronic Engineering, Anhui
University, Hefei 230601, People’s Republic of China  Dong Wang
<EMAIL_ADDRESS>School of Physics & Optoelectronic Engineering, Anhui
University, Hefei 230601, People’s Republic of China

###### Abstract

We studied the dynamics of entropic uncertainty in Markovian and non-Markovian
systems during the charging of open quantum batteries (QBs) mediated by a
common dissipation environment. In the non-Markovian regime, the battery is
almost fully charged efficiently, and the strong non-Markovian property is
beneficial for improving the charging power. In addition, the results show
that the energy storage is closely related to the couplings of the charger-
reservoir and battery-reservoir; that is, the stronger coupling of a charger-
reservoir improves energy storage. In particular, entanglement is required to
obtain the most stored energy and is accompanied by the least tight entropic
bound. Interestingly, it was found that the tightness of the entropic bound
can be considered a good indicator of the energy transfer in different
charging processes, and the complete energy transfer always corresponds to the
tightest entropic bound. Our results provide insight into the optimal charging
efficiency of QBs during practical charging.

##  I Introduction

The concept of quantum battery (QB) was proposed by Alicki and Fannes l7ah1 ,
who investigated the extractability of the amount of work done by a quantum
system during unitary evolution. It is a device used to temporarily store
energy. In recent years, a series of studies have been conducted on energy
changes in quantum batteries (QBs) l7ah1 ; l9 ; l12 ; l8ab12 ; l10ab3ah29 ;
l11ab2 . In practice, both quantum and classical batteries focus on the
efficiency with which energy (in the case of the battery) is transferred
during charging and discharging to extract as much energy as possible from the
battery. QB considered an open system because of its interaction with the
surrounding environment. This leads to the leakage of energy from the QB into
the environment. Thus far, many scholars have proposed various schemes for
studying energy transfer efficiency by quantum effects l14 ; l15 ; l13ab13 .
In principle, the characterizations for improving the performance of the QB
during charging are reflected by higher energy storage and the smaller optimal
charging time (i.e., the shortest time $\lambda t_{s}$ to reach the peak
energy storage) l11ab2 ; l10ab3ah29 ; w1 ; w2 ; b4 ; b6 ; b7 ; b8 . Therefore,
it is necessary to pursue a better charging process by tuning the control
parameters in the transfer process.

In addition, the uncertainty relation was originally proposed by Heisenberg,
where we cannot accurately predict the measurement outcomes with respect to
two arbitrary incompatible observables 1 . Heisenberg’s uncertainty principle
is regarded as a cornerstone of quantum mechanics. Subsequently, Kennard 2
and Robertson 3  proposed an expression of the uncertainty principle in the
form of standard deviation:
$\Delta{\hat{P}_{1}}\cdot\Delta{\hat{P}_{2}}\geq\frac{1}{2}|\langle{[{\hat{P}_{1}},{\hat{P}_{2}}]}\rangle|$,
where $\Delta(\bullet)$ represents the standard deviation, and
$\langle\circ\rangle$ in the lower bound represents the corresponding expected
value. Note that this relation leads to a trivial result, that is, the lower
bound will become zero when the system’s state is prepared in one of the two
observables’ eigenstates. On this premise, Deutsch introduced information
entropy to describe the uncertainty principle and present the entropic
uncertainty relation (EUR) 4 :
$H({\hat{{P_{1}}}})+H({\hat{{P_{2}}}})\geq{\log_{2}}{({\frac{2}{{1+\sqrt{c}}}})^{2}},$
where
$c={\max_{ij}}|\left\langle{{{P_{1i}}}}.\mathrel{\left|{\vphantom{{{P_{1i}}}{{P_{2j}}}}}\right.\kern-1.2pt}{{{P_{2j}}}}\right\rangle{|^{2}}$,
$\left|{{P_{1i}}}\right\rangle$ and $\left|{{P_{2j}}}\right\rangle$ being with
the eigenstates of the measurement operator ${\hat{{P_{1}}}}$
${\hat{{P_{2}}}}$, and the Shannon entropy
$H({{\hat{P}_{k}}})=-\sum\nolimits_{i}{{p_{ki}}}{\log_{2}}{p_{ki}}\left({k=1,2}\right)$
with
${p_{ki}}=\left\langle{{P_{ki}}}\right|{\hat{\rho}}\left|{{P_{ki}}}\right\rangle$
being the probability of obtaining the $i$th measurement result. Later, Kraus
5  and Maassen and Uffink 6  improved Deutsch’s results. Notably, this
relationship is suitable for single-particle systems. Thus, one may ask how to
express the uncertainty relation. If the measured particle is correlated with
another. Berta et al. proposed a new form of uncertainty relation called
quantum memory-assisted entropic uncertainty relation (QMA-EUR) 7 ; m ; x .
Their investigation focuses on a two-measurement case. Later on, Liu $et$
$al.$ expanded Berta et al.’s version to the case of multiple measurements,
and put forward the general expression of a QMA-EUR related to multiple
measurements as liu

$\displaystyle\sum\limits_{x=1}^{N}{S\left({{{\hat{P}}_{x}}\left|B\right.}\right)}\geq-{\log_{2}}\left(b\right)+\left({N-1}\right)S\left({A\left|B\right.}\right),$
(1)

where $S({A|B})=S({{{\hat{\rho}}_{AB}}})-S\left({{{\hat{\rho}}_{B}}}\right)$
is the von Neumann conditional entropy 8 , and the von Neumann entropy
$S\left({\hat{\rho}}\right)=-tr\left({\hat{\rho}{{\log}_{2}}\hat{\rho}}\right)$.
Specifically,
$b=\mathop{\max}\limits_{{i_{N}}}\left\\{{\sum\limits_{{i_{2}}\sim{i_{N-1}}}{\mathop{\max}\limits_{{i_{1}}}}\left[{c\left({u_{{i_{1}}}^{1},u_{{i_{2}}}^{2}}\right)}\right]\prod\limits_{x=2}^{N-1}{\left[{c\left({u_{{i_{x}}}^{x},u_{{i_{x+1}}}^{x+1}}\right)}\right]}}\right\\}$
and $u_{i_{x}}^{x}$ is the $i$th eigenvector of the operator
${{{\hat{P}}_{x}}}$.

Recently, some scholars have studied the change in entanglement in the QB, x18
and the relationship between energy density and entanglement x18+  during
quantum battery charging, which indicates that entanglement plays an important
role in energy transfer during the charging process. In addition, the
connection between entanglement and uncertainty has been disclosed ee . With
this in mind, the tightness of the lower bound on the entropic uncertainty
relation plays a role during the charging of quantum batteries. Motivated by
this, we first investigate the optimization of the charging process by
different coupling regimes: Markovian and non-Markovian regimes. When the
battery is stable, we look for more stored energy by investigating the qubit-
environment coupling and the initial entanglement of the charger and battery.
We found that a stronger charger-environment coupling is beneficial for
obtaining more stored energy. Moreover, we study the variation in the entropic
uncertainty relation (the tightness of the entropic bound) and the relation
with energy storage in charging QBs. Remarkably, it was found that the
tightness of the entropic bound can be considered an indicator of the energy
transfer, and the complete energy transfer is always accompanied by the
tightest entropic bound. That is, the tightness of the uncertainty bound can
reflect the charging efficiency in the QB.

The structure of this paper is as follows. In Section II, charging model of an
open quantum battery system is reviewed. In Section III, the time evolution of
stored energy and power in different coupling regimes is studied in the model.
In Section IV, we investigate the steady-state energy of the QB and its
relation to the tightness of the lower bounds on entropic uncertainty.
Furthermore, we investigate the tightness at different energy transfer rates
(the proportion of energy transferred from the charger to the battery) in the
charging process. Finally, we conclude our study with a concise summary in
Section V.

##  II charging model

We consider a quantum battery charging model consisting of a quantum charger
$A$ and a quantum battery $B$, which is coupled to a common zero-temperature
bosonic reservoir, without coupling between $A$ and $B$. Each cell of $A$ and
$B$ is a two-level system with the same transition frequency,
${\omega_{A}}={\omega_{B}}={\omega_{0}}$, and the ground and excitation states
are $\left|{{g}}\right\rangle$ and $\left|e\right\rangle$, respectively. The
Hamiltonian $H$ of the entire system is composed of ${H_{0}}$ and
${H_{{\mathop{\rm int}}}}$ b23 ; b24 ; b

Figure 1:  (a) Within time $t\in\left[{0,\tau}\right]$, the charger ($A$) and
battery ($B$) interacts with the environment ($E$). (b) When $t>\tau$, at the
end of the charging process, the battery is in the steady-state and its energy
is conserved.

$\displaystyle H={H_{0}}+{H_{{\mathop{\rm int}}}},$ (2)

where

$\displaystyle{H_{0}}=\sum\limits_{i=A,B}{{\omega_{0}}}\sigma_{i}^{+}\sigma_{i}^{-}+\sum\limits_{k}{{\omega_{k}}c_{k}^{\dagger}{c_{k}}},$
(3)

and

$\displaystyle{H_{{\mathop{\rm
int}}}}=\left({{\beta_{1}}\sigma_{A}^{+}+{\beta_{2}}\sigma_{B}^{+}}\right)\sum\limits_{k}{{g_{k}}{c_{k}}}+\left({{\beta_{1}}\sigma_{A}^{-}+{\beta_{2}}\sigma_{B}^{-}}\right)\sum\limits_{k}{{g_{k}}c_{k}^{\dagger}}.$
(4)

${H_{0}}$ represents the free Hamiltonian of the qubit and the reservoir,
where $\sigma_{i}^{+}$ and $\sigma_{i}^{-}$ are the Pauli rising and lowering
operators of the qubit, ${\omega_{k}}$ and $c_{k}^{\dagger}$(${c_{k}}$) are
the frequency and creation(annihilation) operators of the $k$th mode of the
field, respectively. ${H_{{\mathop{\rm int}}}}$ denotes the interaction
between the two-qubit system and reservoir. ${\beta_{1}}$ and ${\beta_{2}}$
are defined as the interaction strengths between qubits $A$ and $B$ and the
reservoir, respectively, and are dimensionless real parameters. And
${g_{k}}{\beta_{1}}$ and ${g_{k}}{\beta_{2}}$ are the coupling constants
between the qubits and reservoir. Then, the collective coupling constant
${\beta_{T}}=\sqrt{\beta_{1}^{2}+\beta_{2}^{2}}$ and the relative interaction
strength
${\zeta_{i}}={{{\beta_{i}}}\mathord{\left/{\vphantom{{{\beta_{i}}}{{\beta_{T}}}}}\right.\kern-1.2pt}{{\beta_{T}}}}\left({i=1,2}\right)$.
It should be noted that different effective couplings of $A$ and $B$ with a
reservoir can be obtained under different conditions for ${{\beta_{1}}}$ and
${{\beta_{2}}}$.

Figure 2: Markovian charging process. The stored energy of battery
$\Delta{E_{B}}\left(t\right)/{\omega_{0}}$ and instantaneous power
$P/{\omega_{0}}$ as function of the dimensionless quantity $\lambda t$ and
coupling strength $R$ are plotted in (a), (b) and (c), (d), respectively. We
choose the initial separable states with $\theta=0.5\pi$, and the same
interactions of charger-reservoir and battery-reservoir
(${\zeta_{1}}={\zeta_{2}}=\frac{1}{{\sqrt{2}}}$). Graphs (a) and (c): $0\leq
R\leq 0.5$; Graphs (b) and (d): $0.5\leq R\leq 1$.

In addition, ${H_{{\mathop{\rm int}}}}$ exists within the time interval:
$t\in\left[{0,\tau}\right]$ of the charging process. Outside this interval,
there is no interaction between $A$($B$) and the environment, furthermore,
there is no interaction between $A$ and $B$. Therefore, this is the wireless
charging process of QB in an environmental medium b . Note that, this charging
model corresponds to physically feasible experimental settings by the all-
optics platform x19 ; x20 ; b .

Under the limitation that the two-qubit system and reservoir are separable
states at the beginning, we consider the case in which there is only one
excitation state in the two-qubit system and the reservoir is in a vacuum
state. The initial state of the entire system is

$\displaystyle\left|{\varphi\left(0\right)}\right\rangle={\eta_{01}}{\left|e\right\rangle_{A}}{\left|{{g}}\right\rangle_{B}}{\left|0\right\rangle_{\rm
X}}+{\eta_{02}}{\left|{{g}}\right\rangle_{A}}{\left|e\right\rangle_{B}}{\left|0\right\rangle_{\rm
X}},$ (5)

where, ${\eta_{01}}$ and ${\eta_{02}}$ are the probabilities amplitudes and
${\left|0\right\rangle_{\rm X}}$ is the vacuum state of the reservoir. The
time evolution of a single excited state can be obtained as

$\displaystyle\left|{\varphi\left(t\right)}\right\rangle$
$\displaystyle={\eta_{1}}\left(t\right){\left|e\right\rangle_{A}}{\left|{{g}}\right\rangle_{B}}{\left|0\right\rangle_{\rm
X}}+{\eta_{2}}\left(t\right){\left|{{g}}\right\rangle_{A}}{\left|e\right\rangle_{B}}{\left|0\right\rangle_{\rm
X}}$
$\displaystyle+\sum\limits_{k}{{\eta_{k}}\left(t\right)}{\left|{{g}}\right\rangle_{A}}{\left|{{g}}\right\rangle_{B}}{\left|1_{k}\right\rangle_{\rm
X}},$ (6)

where, ${\eta_{1}}\left(t\right)$ and ${\eta_{2}}\left(t\right)$ are
probability amplitudes. We consider the case in which the environment is
regarded as an electromagnetic field in a lossless cavity, and the Lorentzian
form of the spectral density of the cavity field is b26

$\displaystyle
J\left(\omega\right)=\frac{{{\nu^{2}}\lambda}}{{\pi{{\left({\omega-{\omega_{0}}}\right)}^{2}}-\pi{\lambda^{2}}}},$
(7)

where, $\lambda$ is the spectrum width and
${1\mathord{\left/{\vphantom{1\lambda}}\right.\kern-1.2pt}\lambda}$ is the
correlation time of the reservoir, and $\nu$ is the effective coupling
strength satisfying $L=\nu{\beta_{T}}$ where $L$ is the vacuum Rabi frequency.
Then we define a dimensionless real number ${R}=L/{\lambda}$, where strong
coupling has ${{R}\gg 1}$ and weak coupling with ${{R}\ll 1}$ b191 ; b192 ;
b193 ; b194 ; b195 ; b196 ; b197 . In this case, Markovian evolution occurs
under weak coupling, and non-Markovian dynamics and memory effects occur under
strong coupling. And then the analytic solution for
${\eta_{i}}\left(t\right)\left({i=1,2}\right)$ comes in the following form b23
; b24 ; b

$\displaystyle{\eta_{1}}\left(t\right)=\left[{\zeta_{2}^{2}+\zeta_{1}^{2}\mu\left(t\right)}\right]{\eta_{01}}-{\zeta_{1}}{\zeta_{2}}\left[{1-\mu\left(t\right)}\right]{\eta_{02}},$
(8)

and

$\displaystyle{\eta_{2}}\left(t\right)=\left[{\zeta_{1}^{2}+\zeta_{2}^{2}\mu\left(t\right)}\right]{\eta_{02}}-{\zeta_{1}}{\zeta_{2}}\left[{1-\mu\left(t\right)}\right]{\eta_{01}},$
(9)

where the Lorentzian form is $\mu\left(t\right)={e^{-\frac{{\lambda
t}}{2}}}\left[{\cosh\left({\frac{{\kappa
t}}{2}}\right)+\frac{\lambda}{\kappa}\sinh\left({\frac{{\kappa
t}}{2}}\right)}\right]$ and $\kappa=\sqrt{{\lambda^{2}}-4{L^{2}}}$ b23 ; b24 ;
b . Finally, we have

$\displaystyle\mu\left(t\right)=\left\\{{\begin{array}[]{*{20}{c}}{{e^{-\frac{{\lambda
t}}{2}}}\left[{\cosh\left(x\right)+\sinh\left(x\right)/{\chi_{1}}}\right],}&{0.5\geq
R\geq 0}\\\ {{e^{-\frac{{\lambda
t}}{2}}}\left[{\cos\left(y\right)+\sin\left(y\right)/{\chi_{2}}}\right],}&{R\geq
0.5}\end{array}}\right.$ (12)

where $x={\chi_{1}}\sqrt{{\lambda^{2}}{t^{2}}}/2$,
$y={\chi_{2}}\sqrt{{\lambda^{2}}{t^{2}}}/2$ and
${\chi_{1}}=\sqrt{1-4{R^{2}}}$, ${\chi_{2}}=\sqrt{4{R^{2}}-1}$.

##  III Performance of quantum batteries with different coupling regimes

During the charging process of the quantum batteries, the coupling strength
between the qubits (charger and battery) and the reservoir is considered to be
an important element in determining excellent charging performance. From Eqs.
(5) and (6), the reduced density matrices can be expressed as:

$\displaystyle\rho_{AB}{\left(0\right)}$
$\displaystyle=|{\eta_{01}}{|^{2}}{\left|{e{{g}}}\right\rangle_{AB}}\left\langle{e{{g}}}\right|+|{\eta_{02}}{|^{2}}{\left|{{{g}}e}\right\rangle_{AB}}\left\langle{{{g}}e}\right|$
$\displaystyle+{\eta_{01}}\eta_{02}^{*}{\left|{e{{g}}}\right\rangle_{AB}}\left\langle{{{g}}e}\right|+\eta_{01}^{*}{\eta_{02}}{\left|{{{g}}e}\right\rangle_{AB}}\left\langle{e{{g}}}\right|,$
(13)

and

$\displaystyle{\rho_{AB}}\left(t\right)$
$\displaystyle=|{\eta_{1}}\left(t\right){|^{2}}{\left|{eg}\right\rangle_{AB}}\langle
eg|+|{\eta_{2}}\left(t\right){|^{2}}{\left|{ge}\right\rangle_{AB}}\langle ge|$
$\displaystyle+{\eta_{1}}\left(t\right){\eta_{2}}{\left(t\right)^{*}}{\left|{eg}\right\rangle_{AB}}\langle
ge|+{\eta_{1}}{\left(t\right)^{*}}{\eta_{2}}\left(t\right){\left|{ge}\right\rangle_{AB}}\langle
eg|$
$\displaystyle+\left[{1-|{\eta_{1}}\left(t\right){|^{2}}-|{\eta_{2}}\left(t\right){|^{2}}}\right]{\left|{gg}\right\rangle_{AB}}\langle
gg|$ (14)

of the two-qubit system. Note that $\eta_{i}(t)=\eta_{i}(t)^{*},(i=1,2)$. The
reduced-density matrices $\rho_{A}{\left(t\right)}$ and
$\rho_{B}{\left(t\right)}$ of $A$ and $B$ can also be obtained. Then we take
${\eta_{01}}=\sin\theta\cdot{e^{i\phi}}$ and ${\eta_{02}}=\cos\theta$.
According to Eq. (14) and the reduced density matrix
$\rho_{B}{\left(t\right)}$, the energy of the battery at time $t$ is
represented by
${{E}_{B}}\left(t\right)=tr\left[{{H_{B}}\rho_{B}{{\left(t\right)}}}\right]$.
Hence, the change in battery energy during charging can be quantified by

$\displaystyle\Delta{E_{B}}\left(t\right)$
$\displaystyle={E_{B}}\left(t\right)-{E_{B}}\left(0\right)$
$\displaystyle={\omega_{0}}\left[{|{\eta_{2}}\left(t\right){|^{2}}-\cos^{2}{\theta}}\right].$
(15)

Another effective parameter in the charging process is the charging power of
the QB. In this case, the instantaneous power of the charging process is:

$\displaystyle{P}=\frac{{{E_{B}}(t)}}{t}.$ (16)

###  III.1 Markovian dynamics

Figure 3: In the Non-Markovian charging process the stored energy of the
battery $\Delta{E_{B}}\left(t\right)/{\omega_{0}}$ and instantaneous power
$P/{\omega_{0}}$ as function of the dimensionless quantity $\lambda t$ and
coupling strength $R$ are plotted in Graphs (a) and (b), respectively. $1\leq
R\leq 500$ is set and all other parameters are the same as those in Fig. 2.

In the weakly coupled regime, the general behaviour of energy variation can be
obtained. We consider the maximum energy storage and the faster charging speed
(i.e., greater instantaneous power) as indicators to optimize the charging
process and find the smallest optimal charging time ${\lambda t_{e}}$ when the
maximum energy obtained by the battery is fixed.

Fig. 2(a) shows the effect of coupling strength on stored energy with
${\zeta_{1}}={\zeta_{2}}=\frac{1}{{\sqrt{2}}}$. First, the stored energy
increases to its maximum over time. Second, in the long-time limit, the change
in the coupling between the qubit and the reservoir cannot increase the peak
energy storage; however, it can significantly optimize the shortest time
${\lambda t_{s}}$ to reach the peak energy storage. Fig. 2(b) represents the
variation in the energy stored over a larger range of coupling. The energy did
not change monotonically with time. Different from Fig. 2(a), the enhanced
coupling strength enables the QB to store more energy with smaller ${\lambda
t_{s}}$. Fig. 2(c) reveals the contribution of coupling strength to
instantaneous power during charging. A stronger interaction between the two-
qubit system and the reservoir was found to speed up the charging. Compared
with Fig. 2(c) and Fig. 2(d) also reveals that the improvement of the coupling
strength is beneficial to the charging power, which is increased by about 4
times. Moreover, the change trend of power does not change like that of
energy, that is, it reaches the maximum in a relatively short period of time
and then decreases slowly. In short, in Markovian systems, a higher coupling
strength reduces the optimal charging time and attains more stored energy;
therefore, the charging performance of the battery is improved.

###  III.2 Non-Markovian dynamics

Fig. 3(a) clearly shows the help of a memory effect for maintaining high
energy storage. The stored energy oscillates efficiently, and the first wave
packet represents the maximum stored energy, which corresponds to the optimal
charging time. The peaks of the other wave packets decreased with respect to
${\lambda t}$. It can be seen that changing the coupling strength does not
significantly improve the maximum stored energy, and a decrease in ${\lambda
t_{s}}$ can be realized by an increase in the coupling. Fig. 3(b) illustrates
that coupling enhancement can significantly increase the charging speed, which
is the same as Fig. 2(c) and (d).

In the Markovian regime, one can obtain that the maximum energy of the weak
coupling regime is about $0.34{\omega_{0}}$ and the optimal charging time is
about 4. However, the non-Markovian system has advantages both in terms of
maximum energy storage (the battery is almost full) and optimal charging time
$\lambda{t_{e}}\approx 0.006$, owing to the unique memory effect of the non-
Markovian regime, which provides an important way to overcome energy leakage
into the environment.

##  IV the tightness of an entropic bound and stored energy

In the open quantum battery system, the decoherence effect of the environment
leads to an energy leakage. Although greater non-Markovian properties allow
the battery to charge satisfactorily for short periods of time, there still
exists an additional improvement in energy storage for a longer charging time,
which is regarded as another consideration for an excellent charging protocol
for quantum batteries. Therefore, in this section, we pursue higher stored
energy in the steady state of a quantum battery under the influence of initial
entanglement and relative interaction strength ( characterized by comparing
the charger-reservoir $\zeta_{1}$ and battery-reservoir coupling strength
$\zeta_{2}$. Note that $\zeta_{1}$ and $\zeta_{2}$ were normalized).
Furthermore, we investigate the role of the entanglement and tightness of the
entropic bound in the process of boosting stored energy.

For a system with an X-type density matrix, the concurrence of the system
wootters  is
$C_{\rho(t)}=2\max\\{0,|{\rho_{14}}|-\sqrt{{\rho_{22}}{\rho_{33}}},|{\rho_{23}}|-\sqrt{{\rho_{11}}{\rho_{44}}}\\}$,
where ${\rho_{ij}}$ corresponds to the entries in row $i$ and column $j$ of
matrix $\rho(t)$. Consequently, the concurrence of $\rho_{AB}{\left(t\right)}$
is calculated as ${C_{{\rho_{AB}}(t)}}=2|{\eta_{1}}(t){\eta_{2}}^{*}(t)|$.

To probe the dynamics of entropic uncertainty, we resort to a pair of Pauli
operators ${\sigma_{x}}$ and ${\sigma_{z}}$ as the incompatibility, which
leads to post-measurement states

$\displaystyle\rho\left({{\sigma_{x}}|B}\right)=\frac{1}{2}\left[{\rho{{\left(t\right)}_{AB}}+\left({{\sigma_{x}}\otimes
I}\right)\rho{{\left(t\right)}_{AB}}\left({{\sigma_{x}}\otimes
I}\right)}\right],$ (17)

and

$\displaystyle\rho\left({{\sigma_{z}}|B}\right)=\frac{1}{2}\left[{\rho{{\left(t\right)}_{AB}}+\left({{\sigma_{z}}\otimes
I}\right)\rho{{\left(t\right)}_{AB}}\left({{\sigma_{z}}\otimes
I}\right)}\right].$ (18)

where $I$ denotes an identity matrix. With respect to
$\rho\left({{\sigma_{x}}|B}\right)$, eigenvalues can be calculated as

$\displaystyle\begin{array}[]{*{20}{c}}{\gamma_{1}^{x}=\gamma_{2}^{x}=\left({1-D}\right)/4,}\\\
{\gamma_{3}^{x}=\gamma_{4}^{x}=\left({1+D}\right)/4,}\end{array}$ (21)

where
$D={\sqrt{1-4|{\eta_{2}}\left(t\right){|^{2}}+4|{\eta_{1}}\left(t\right){|^{2}}|{\eta_{2}}\left(t\right){|^{2}}+4|{\eta_{2}}\left(t\right){|^{4}}}}$,
for $\rho\left({{\sigma_{z}}|B}\right)$, we have eigenvalues

$\displaystyle\begin{array}[]{c}\gamma_{1}^{z}=0,\gamma_{2}^{z}=|{\eta_{1}}\left(t\right){|^{2}},\gamma_{3}^{z}=|{\eta_{2}}\left(t\right){|^{2}},\\\
\gamma_{4}^{z}=1-|{\eta_{1}}\left(t\right){|^{2}}-|{\eta_{2}}\left(t\right){|^{2}}.\end{array}$
(24)

Finally, we obtain the magnitude of the entropic uncertainty as:
$U_{l}^{xz}\left(t\right)=2\sum\limits_{i=1}^{2}{\varepsilon_{i}^{B}}{\log_{2}}\varepsilon_{i}^{B}-\sum\limits_{j=1}^{4}{\gamma_{j}^{x}}{\log_{2}}\gamma_{j}^{x}-\sum\limits_{j=1}^{4}{\gamma_{j}^{z}}{\log_{2}}\gamma_{j}^{z}$
(left-hand side of Eq. (1)). When the system is stable, we have

$\displaystyle U_{l}^{xz}\left(\infty\right)$
$\displaystyle={\left|{{\eta_{2}}\left(\infty\right)}\right|^{2}}{\log_{2}}{\left|{{\eta_{2}}\left(\infty\right)}\right|^{2}}-M{\log_{2}}M$
$\displaystyle-2\left[{{{\left|{{\eta_{2}}\left(\infty\right)}\right|}^{2}}-1}\right]{\log_{2}}\left[{1-{{\left|{{\eta_{2}}\left(\infty\right)}\right|}^{2}}}\right]$
$\displaystyle-{\left|{{\eta_{1}}\left(\infty\right)}\right|^{2}}{\log_{2}}{\left|{{\eta_{1}}\left(\infty\right)}\right|^{2}}$
$\displaystyle+\left({F-1}\right){\log_{2}}\left[{\left({1-F}\right)/4}\right]/2$
$\displaystyle-\left({F+1}\right){\log_{2}}\left[{\left({F+1}\right)/4}\right]/2,$
(25)

and lower bound
$U_{r}^{2}\left(t\right)=1+\sum\limits_{i=1}^{2}{\varepsilon_{i}^{B}}{\log_{2}}\varepsilon_{i}^{B}-\sum\limits_{j=1}^{4}{\varepsilon_{j}^{AB}}{\log_{2}}\varepsilon_{j}^{AB}$
( right-hand side of Eq. (1)). The corresponding lower bound in the steady
state is in the form of

$\displaystyle U_{r}^{2}\left(\infty\right)$
$\displaystyle=1+{\left|{{\eta_{2}}\left(\infty\right)}\right|^{2}}{\log_{2}}{\left|{{\eta_{2}}\left(\infty\right)}\right|^{2}}$
$\displaystyle-\left[{{{\left|{{\eta_{2}}\left(\infty\right)}\right|}^{2}}-1}\right]{\log_{2}}\left[{1-{{\left|{{\eta_{2}}\left(\infty\right)}\right|}^{2}}}\right]$
$\displaystyle-M{\log_{2}}M-\left({1-M}\right){\log_{2}}\left({1-M}\right),$
(26)

where ${\varepsilon_{i}^{B}}$ and ${\varepsilon_{j}^{AB}}$ are the eigenvalues
of $\rho_{B}{\left(t\right)}$ and $\rho_{AB}{\left(t\right)}$,
$F=\sqrt{1+4{{\left|{{\eta_{1}}\left(\infty\right)}\right|}^{2}}{{\left|{{\eta_{2}}\left(\infty\right)}\right|}^{2}}-4{{\left|{{\eta_{2}}\left(\infty\right)}\right|}^{2}}+4{{\left|{{\eta_{2}}\left(\infty\right)}\right|}^{4}}}$,
and
$M=1-{\left|{{\eta_{1}}\left(\infty\right)}\right|^{2}}-{\left|{{\eta_{2}}\left(\infty\right)}\right|^{2}}$.
The superscripts of entropic uncertainty and lower bounds indicate the Pauli
operator chosen for the measurement and the number of measurement operators,
respectively.

Figure 4: When the battery is in the steady-state, the concurrence
${C_{{\rho_{AB}}(\infty)}}$, stored energy,
$\Delta{E_{B}}(\infty)/{\omega_{0}}$ and the tightness of the lower bound
$\Delta U^{xz}(\infty)$ are affected by the relative interaction strength
${\zeta_{i}}$ and the probability amplitude of the initial state.
${C_{{\rho_{AB}}(\infty)}}$ is plotted in (a) and (d);
$\Delta{E_{B}}(\infty)/{\omega_{0}}$ is plotted in (b) and (e); $\Delta
U^{xz}(\infty)$ is plotted in (c) and (f); $\theta\in[0.25\pi,0.5\pi]$ and
${\zeta_{1}}\in[0,1]$ are given, respectively. In addition, the relative phase
is also associated with the concurrence ${C_{{\rho_{AB}}(\infty)}}$, stored
energy, $\Delta{E_{B}}(\infty)/{\omega_{0}}$ and the tightness of the lower
bound $\Delta U^{xz}(\infty)$, here, we set $\phi=0$ in Graphs (a), (b), and
(c); and $\phi=\pi$ in Graphs (d), (e) and (f).

After a sufficient charging period, i.e., $t\to\infty$, the internal energy of
the battery tends to be stable, and at this time we get $\mu(\infty)=0$,
${\eta_{1}}(\infty)=\zeta_{2}^{2}{\eta_{01}}-{\zeta_{1}}{\zeta_{2}}{\eta_{02}}$,
and
${\eta_{2}}(\infty)=\zeta_{1}^{2}{\eta_{02}}-{\zeta_{1}}{\zeta_{2}}{\eta_{01}}$.
Thus, the The concurrence can be expressed as follows:

$\displaystyle{C_{{\rho_{AB}}(\infty)}}=2|{\eta_{1}}(\infty)\eta_{2}^{*}(\infty)|,$
(27)

and the stored energy of the battery in the steady state is obtained by:

$\displaystyle\Delta{E_{B}}\left(\infty\right)={\omega_{0}}\left[|{\eta_{2}}\left(\infty\right){|^{2}}-\cos^{2}\theta\right].$
(28)

Similarly, entropic uncertainty ${U_{l}}^{xz}(\infty)$ and its lower bound
${U_{r}^{2}}(\infty)$ in the steady state can be calculated. The tightness of
the lower bound is given by $\Delta
U^{xz}(\infty)={U_{l}}^{xz}(\infty)-{U_{r}^{2}}(\infty)$, which reflects the
general characteristics of the entropic uncertainty relation. The tighter
lower bound is quantified by a smaller $\Delta U^{xz}(\infty)$.

In addition, as a parameter in the probability amplitude of the initial state,
$\theta$ is reflected in the initial charger-battery entanglement in the
charging scheme. The concurrent initial state can be expressed as
${C_{{\rho_{AB}}(0)}}=2|{\eta_{01}}\eta_{02}^{*}|$. After the calculation, it
was found that the entanglement of the initial state is only related to
$\theta$ and is independent of the phase $\phi$, which reaches a maximum value
of $1$ when $\theta=0.25\pi$ and monotonically decreases to $0$ when $\theta$
approaches the boundary value $0.5\pi$. Therefore, in the charging process, we
consider the influence of the initial entanglement on battery performance
according to the change in $\theta$.

In Fig. 4(a), (b), and (c), entanglement and stored energy have a similar
evolution process while they are opposite to the tightness of entropic bound.
All three quantities reach an extreme value when the initial entanglement is
at a minimum. Moreover, when the initial state is separable ($\theta=0.5\pi$),
the stored energy is effective ($\Delta{E_{B}}(\infty)\geq 0$) for all values
of the relative interaction strength, and the stored energy reaches a maximum
($\Delta{E_{B}}{(\infty)_{\max}}=0.25{\omega_{0}}$) when the coupling
strengths of the charger-reservoir and the battery-reservoir are symmetrical
(${\zeta_{1}}={\zeta_{2}}={1\mathord{\left/{\vphantom{1{\sqrt{2}}}}\right.\kern-1.2pt}{\sqrt{2}}}$).
The results also show that entanglement is beneficial for energy storage in
when $\phi=0$. Although a tighter entropic bound is not required for the
growth of stored energy, there is a close relationship between them, that is,
the highest stored energy strictly corresponds to the least tight entropic
bound, which never occurs with respect to entanglement.

In Fig. 4(d), (e), and (f), with the reversal of the relative phase to
$\phi=\pi$, it can be seen that entanglement reaches the peak value when the
system is initially the maximum entangled state, which is completely opposite
to that of the case of $\phi=0$. Simultaneously, the entropic bound became
less tight. Furthermore, there is effective stored energy
($\Delta{E_{B}}(\infty)\geq 0$), regardless of the initial charger-battery
entanglement. Compared to the case of $\phi=0$, the peak value of the stored
energy increases to approximately $0.33{\omega_{0}}$ when $\phi=\pi$,
accompanied by an asymmetric charging process (${\zeta_{1}}\approx 0.8$) and a
certain initial entanglement ($\theta\approx 1.3$). Comparing Fig. 4(d) and
(e), it is find that charger-battery entanglement is necessary in the
parameter plane $\theta-{\zeta_{1}}$ to obtain the most stored energy.
Interestingly, the flip of the relative phase does not change the close
relationship between the tightness of the entropic bound and the stored
energy, that is, the most stored energy when $\phi=\pi$ strictly corresponds
to the least tight entropic bound.

Figure 5: The concurrence ${C_{{\rho_{AB}}(t)}}$ and the tightness $\Delta
U^{xz}(t)$ as functions of energy are plotted in (a) and (b), respectively.
The x-axis is the energy change of the charger
$\Delta{E_{A}}(t)/{\omega_{0}}$, and the y-axis is the stored energy of the
battery $\Delta{E_{B}}(t)/{\omega_{0}}$. In addition, a separable initial
state $\theta=0.5\pi$ is set.

We find that the stored energy of the steady state is always negatively
correlated with the tightness of the entropic bound when the relative phase is
$0$ or $\pi$. Therefore, we will further study the relationship between the
stored energy and tightness of the entropic bound from the transfer of energy
between charger and battery. Here, we study the evolution of the entanglement
and tightness of the entropic bound for any case of energy transfer. The
energy variation of the charger and battery is associated with the tightness
of the lower bound and the entanglement (according to Eqs. (25), (26), (27),
and (28)). From the relation
${\left|{{\eta_{1}}\left(t\right)}\right|^{2}}=\Delta{E_{A}}(t)/{\omega_{0}}+\sin^{2}{\theta}$
and
${\left|{{\eta_{2}}\left(t\right)}\right|^{2}}=\Delta{E_{B}}(t)/{\omega_{0}}+\cos^{2}{\theta}$,
where, $\left|{\Delta{E_{A}}\left(t\right)}\right|$ is the energy lost by the
charger, we can see that the energy partly determines the evolutionary
behavior of entanglement or tightness. The concurrence can then be expressed
as
${C_{{\rho_{AB}}\left(t\right)}}=2{\left[{\Delta{E_{A}}\left(t\right)/{\omega_{0}}+\sin^{2}{{\theta}}}\right]^{\frac{1}{2}}}{\left[{\Delta{E_{B}}\left(t\right)/{\omega_{0}}+\cos^{2}{{\theta}}}\right]^{\frac{1}{2}}}$.
Similarly, the tightness of the entropic bound can be represented by
$\Delta{U^{xz}}\left(t\right)$, which is obtained by substituting energy and
parameters $\theta$.

Next, we visually depict the relationship between entanglement or tightness
and energy storage in Fig. 5. The energy stored by the battery during the
charging process has a boundary value, which is the maximum value of the
stored energy regardless of the energy lost by the charger, as shown in Fig. 5
as the hypotenuse of a triangle region. The boundary value or constraint can
be expressed as $\Delta{E_{B}}(t)/|\Delta{E_{A}}(t)|=1$. This implies that the
energy lost by the charger is completely transferred to the battery. However,
we find that at $\Delta{E_{B}}(t)\neq|\Delta{E_{A}}(t)|$, that is,
$\Delta{E_{B}}\left(t\right)/|\Delta{E_{A}}\left(t\right)|<1$ the energy is
not completely transferred to the battery, which means that some of the
transferred energy flows into the environment. Different energy transfers
actually correspond to arbitrary charging processes caused by different
parameter settings in the charging model. Therefore, it is meaningful to
discuss the effect of energy transfer rate
$\Delta{E_{B}}\left(t\right)/|\Delta{E_{A}}\left(t\right)|$ on the tightness
of entropic bound.

Meanwhile, Fig. 5(a) shows that the relationship between entanglement and
energy variation during the arbitrary charging process. It can be seen that an
increase in the energy transfer rate (i.e., the charger loses less energy or
the battery stores more energy) contributes to an increase in entanglement.
Entanglement disappears when the battery or charger is empty. Interestingly,
the energy can be completely transferred for different entanglements,
corresponding to the states of the hypotenuse in Fig. 5(a). Therefore, we can
conclude that entanglement is not a clear indicator of the maximal energy
stored. Besides, we obtain that the maximum energy transfer rate corresponds
to the tightest entropic bound except when the charger or battery is
completely empty, as shown in Fig. 5(b). This indicates that the tightness of
the entropic bound can also play a role in indicating the degree of energy
transfer in the charging processes, implying that the energy is completely
transferred for the tightest entropic bound $\Delta{U^{xz}}\left(t\right)=0$,
which virtually agrees with the previous conclusion.

To explore the role of tightness in improving charging in a more general
context. We consider three measurements with
$(\sigma_{x},\sigma_{y},\sigma_{z})$. As a result, the entropic uncertainty is

$\displaystyle U_{l}^{xyz}\left(t\right)$
$\displaystyle=-{\left|{{\eta_{1}}\left(t\right)}\right|^{2}}{\log_{2}}{\left|{{\eta_{1}}\left(t\right)}\right|^{2}}+2{\left|{{\eta_{2}}\left(t\right)}\right|^{2}}{\log_{2}}{\left|{{\eta_{2}}\left(t\right)}\right|^{2}}$
$\displaystyle-3\left[{{{\left|{{\eta_{2}}\left(t\right)}\right|}^{2}}-1}\right]{\log_{2}}\left[{1-{{\left|{{\eta_{2}}\left(t\right)}\right|}^{2}}}\right]-m{\log_{2}}m$
$\displaystyle+\left({D-1}\right){\log_{2}}\left[{\left({1-D}\right)/4}\right]$
$\displaystyle-\left({D+1}\right){\log_{2}}\left[{\left({1+D}\right)/4}\right],$
(29)

where
$m=1-{\left|{{\eta_{1}}\left(t\right)}\right|^{2}}-{\left|{{\eta_{2}}\left(t\right)}\right|^{2}}$
and the corresponding lower bound is

$\displaystyle{U_{r}^{3}}\left(t\right)=2{U_{r}^{2}}\left(t\right)-1,$ (30)

the tightness of the three measurements was $\Delta
U^{xyz}(t)=U_{l}^{xyz}(t)-U_{r}^{3}(t)$.

Figure 6: Tightness $\Delta U^{xyz}(t)$ as functions of energy are plotted.
The other settings are the same as those in Fig. 5.

Fig. 6 shows the connection between the exchanged energy and the tightness of
the entropic uncertainty. Obviously, the complete energy transfer takes place
when the tightness is fixed to $\Delta{U^{xyz}}\left(t\right)=1$ corresponding
to the hypotenuse, which also supports tightness can be considered an
effective indicator of the maximum energy transfer during the charging of QBs.

##  V Discussions and conclusions

In summary, we have investigated the dynamics of the entropic uncertainty of
charging quantum batteries in common dissipative bosonic environments. We
studied two cases of the charging process, and the results show that the
coupling enhancement during the Markovian charging process cannot only improve
the charging power, but also increase the stored energy of the battery.
Stronger coupling in non-Markovian systems can also significantly improve the
charging power; in contrast, the battery can be efficiently full during the
non-Markovian charging process. When the battery is in a steady state, maximum
energy storage is achieved when the charger is more coupled to the reservoir
than the battery, in the case of a certain initial charger-battery
entanglement. It is worth mentioning that a tighter entropic bound has a
negative effect on energy storage in the presence of entanglement. We then
studied the dynamic behavior of the entanglement and tightness in an arbitrary
charging process, and the results revealed that an increase in entanglement
can increase the energy transfer rate and degrade the energy storage.
Remarkably, it is argued that the tightness of the entropic bound can be
regarded as an important indicator of the optimal energy transfer during
charging, and the tightest entropic bound corresponds to the complete energy
transfer. We believe that our findings will be helpful for the pursuit of
high-performance energy transfer in $N$-cell QB charging in the future.

###### Acknowledgements.

This study was supported by the National Natural Science Foundation of China
(Grant Nos. 12075001, 61601002, and 12175001). Anhui Provincial Key Research
and Development Plan (Grant No. 2022b13020004), Anhui Provincial Natural
Science Foundation (Grant No. 1508085QF139), and Fund of the CAS Key
Laboratory of Quantum Information (Grant No. KQI201701).

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