#  $\eta-$Ricci solitons on contact pseudo-metric manifolds

Eftekhar Asgharzadeh and Morteza Faghfouri
Email:ef.asgharzadeh<EMAIL_ADDRESS>of Pure
Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz,
Iran.

###### Abstract

In this paper, we prove that a Sasakian pseudo-metric manifold which admits an
$\eta-$Ricci soliton is an $\eta-$Einstein manifold, and if the potential
vector field of the $\eta-$Ricci soliton is not a Killing vector field then
the manifold is $\mathcal{D}-$homothetically fixed, and the vector field
leaves the structure tensor field invariant. Next, we prove that a $K-$contact
pseudo-metric manifold with a gradient $\eta-$Ricci soliton metric is
$\eta-$Einstein. Moreover, we study contact pseudo-metric manifolds admitting
an $\eta-$Ricci soliton with a potential vector field point wise colinear with
the Reeb vector field. Finally, we study gradient $\eta-$Ricci solitons on
$(\kappa,\mu)$-contact pseudo-metric manifolds.

##  1 Introduction

A Ricci soliton is a natural generalization of an Einstein metric, which was
introduced by Hamilton [17] as the fixed point of the Hamilton’s Ricci flow
$\frac{\partial}{\partial t}g=-2\operatorname{Ric}$. The Ricci flow is a
nonlinear diffusion equation analogue of the heat equation for metrics. A
Ricci soliton $(g,V,\lambda)$ on the pseudo-Riemannian manifold $(M,g)$ is
defined by the following equation

$\mathsterling_{V}g+2\operatorname{Ric}+2\lambda g=0,$

where $\mathsterling_{V}$ is the Lie derivative along the potential vector
field $V$, and $\lambda$ is a constant real number. The Ricci soliton is
called shrinking, steady, expanding if $\lambda<0$, $\lambda=0$ and
$\lambda>0$, respectively. If $V=Df$, where $Df$ is the gradient of the smooth
function $f$, then the Ricci soliton is called a gradient Ricci soliton. Ricci
solitons have been studied in many different contexts (see [10, 2, 11, 25, 14,
22, 3, 16]). Also, they are interests of physicists because of their relations
to string theory [1, 20], and physicists refer to Ricci solitons as quasi-
Einstein metrics [13].

The $\eta-$Ricci soliton notion, as a generalization of a Ricci soliton, was
introduced by Cho and Kimura [9]. An $\eta-$Ricci soliton on a manifold $M$ is
a tuple $(g,V,\lambda,\mu)$, where $g$ is a pseudo-Riemannian metric, $V$ is
the potential vector field, and $\lambda,\mu$ are constant real numbers
satisfying

$\mathsterling_{V}g+2\operatorname{Ric}+2\lambda g+2\mu\eta\otimes\eta=0,$ (1)

where $\eta$ is a $1-$form on $M$. Moreover, if $V=Df$, the $\eta-$Ricci
soliton is called a gradient $\eta-$Ricci soliton and Eq.(1) becomes

$\operatorname{Hess}f+\operatorname{Ric}+\lambda g+\mu\eta\otimes\eta=0.$ (2)

The $\eta-$Ricci solitons have been studied in many different settings, Blaga
studied $\eta-$Ricci solitons on para-Kenmotsu [4] and Lorentzian para-
Sasakian manifolds [5]. Devaraja and Venkatesha studied $\eta-$Ricci solitons
on para-Sasakian manifolds [23], etc.

Contact geometry is an odd-dimensional analogue of the symplectic geometry and
has been studied in many different contexts (particularly) those related to
physics. It has been used as a proper framework for classical thermodynamics
[6, 27], and as a geometrical approach to magnetic field [7]. Also, it was
studied in relation with the Yang-Mills theory [19], quantum mechanics [18],
gravitational waves [21], etc. Studying contact structures with pseudo-
Riemannian metrics was started by Takahashi in [26], but he just studied the
Sasakian case. Recently, Calvaruso and Perrone [8] have studied a contact
pseudo-metric manifold in the general case. Ghaffarzadeh and second author
studied nullity conditions on the contact pseudo-metric manifolds and have
introduced the “$(\kappa,\mu)-$contact pseudo-metric manifold” notion [15].
The relevance for the general relativity of contact pseudo-metric manifolds
was studied in [12]. All of these applications have motivated us to study
$\eta-$Ricci solitons in the contact pseudo-Riemannian settings.

The present paper has been organized as follows. In Section 2, we recalled the
contact pseudo-metric manifold notion and proved some lemmas that are used in
the next sections. In Section 3, we studied $\eta-$Ricci solitons on Sasakian
pseudo-metric manifolds and showed that a Sasakian pseudo-metric manifold,
which admits an $\eta-$Ricci soliton, is an $\eta-$Einstein manifold and if
the potential vector field of the $\eta-$Ricci soliton is not a Killing vector
field, then the manifold is $\mathcal{D}-$homothetically fixed, and presented
an example for it. Moreover, we showed a $K-$contact pseudo-metric manifold
which admits a gradient $\eta-$Ricci soliton is an $\eta-$Einstein manifold.
Also, we studied an $\eta-$Ricci soliton that has a potential vector field
colinear to the Reeb vector field on a contact pseudo-metric manifold and
showed that the manifold is $K-$contact. In the last section, we studied
gradient $\eta-$Ricci solitons on a $(\kappa,\mu)-$contact pseudo-metric
manifold and obtained some conditions on the curvature tensor of the manifold.

##  2 Preliminaries

In this section, we recall some definitions and results needed in the rest of
the paper.

A $(2n+1)-$dimensional manifold $M$ is called an almost contact pseudo-metric
manifold, if there exists an almost contact pseudo-metric structure
$(\varphi,\xi,\eta,g)$ on $M$, where $\varphi,\xi,\eta,g$ are a $(1,1)-$tensor
field, a vector field, a $1$-form and a compatible pseudo-Riemannian metric,
respectively, which satisfy the following equations

$\displaystyle\eta(\xi)=1,\quad\varphi^{2}(X)=-X+\eta(X)\xi,$ (3)
$\displaystyle g(\varphi X,\varphi Y)=g(X,Y)-\epsilon\eta(X)\eta(Y),$ (4)

where $\epsilon=\pm 1$, and $X,Y$ are arbitrary vector fields. Using the above
equations, we have

$\displaystyle\varphi\xi=0,\quad\eta\circ\varphi=0,$
$\displaystyle\eta(X)=\epsilon g(\xi,X),\quad g(\varphi X,Y)=-g(X,\varphi Y)$

and especially $g(\xi,\xi)=\epsilon$. Notice that the signature of the metric
$g$ is $(2p+1,2n-2p)$ if $\xi$ is a spacelike vector field $(g(\xi,\xi)>0)$,
and is $(2p,2n-2p+1)$ if $\xi$ is a timelike vector field ($g(\xi,\xi)<0$).

The fundamental $2-$form $\Phi$ of an almost contact pseudo-metric manifold
$(M,\varphi,\xi,\eta,g)$ is defined as $\Phi(X,Y)=g(X,\varphi Y)$, where
$X,Y\in\Gamma(M)$. If

$g(X,\varphi Y)=(d\eta)(X,Y),$ (5)

then $\eta$ is a contact form, $(\varphi,\xi,\eta,g)$ is a contact pseudo-
metric structure and $M$ is called a contact pseudo-metric manifold.

Throughout this paper, we use $R(X,Y)=[\nabla_{X},\nabla_{Y}]-\nabla_{[X,Y]}$,
where $X,Y\in\Gamma(M),$ as the Riemannian curvature tensor definition. In a
contact pseudo-metric manifold $(M,\varphi,\xi,\eta,g)$ the $(1,1)-$tensor
field $\ell$ and $h$ are defined by

$\ell X=R(X,\xi)\xi,\quad hX=\frac{1}{2}(\mathsterling_{\xi}\varphi)X.$

Also, notice the $\ell$ and $h$ are self-adjoint operators. In the contact
pseudo-metric manifold $(M,\varphi,\xi,\eta,g)$, we have the following
equations [8, 24]

$\displaystyle\operatorname{trace}(h)=\operatorname{trace}(h\varphi)=0,$ (6)
$\displaystyle\eta\circ h=0,\quad\ell\xi=0,$ (7) $\displaystyle
h\varphi=-\varphi h,\quad h\xi=0,$ (8)
$\displaystyle\nabla_{\xi}\varphi=0,\quad$ (9)
$\displaystyle\nabla_{X}\xi=-\epsilon\varphi X-\varphi hX,$ (10)
$\displaystyle\operatorname{Ric}(\xi,\xi)=2n-trh^{2},$ (11)

where $X$ is an arbitrary vector field.

A contact pseudo-metric manifold $(M,\varphi,\xi,\eta,g)$ is a $K-$contact
pseudo-metric manifold if $\xi$ is a Killing vector field or equivalently
$h=0$. So, we have the following equations

$\displaystyle Q\xi=2n\epsilon\xi,$ (12)
$\displaystyle\nabla_{X}\xi=-\epsilon\varphi X,$ (13)

where $Q$ is the Ricci operator of the metric $g$ and $X\in\Gamma(M).$

######  Lemma 2.1.

Let $(M,\varphi,\xi,\eta,g)$ be a $(2n+1)-$dimensional $K-$contact pseudo-
metric manifold, then

$\displaystyle(\nabla_{X}Q)\xi=-2n\varphi X+\epsilon Q\varphi X,$ (14)
$\displaystyle(\nabla_{\xi}Q)X=\epsilon(Q\varphi-\varphi Q)X,$ (15)

where $X$ is an arbitrary vector field.

###### Proof.

First, differentiating (12) along an arbitrary vector field $X$ and using
(13), we obtain (14). Next Lie differentiating $\operatorname{Ric}$, along
$\xi$, we find

$(\mathsterling_{\xi}\operatorname{Ric})(X,Y)=g((\nabla_{\xi}Q)X+Q(\nabla_{x}\xi),Y)+g(QX,\nabla_{Y}\xi),$

where $X,Y\in\Gamma(M)$. Because $\xi$ is a Killing vector field, so
$\mathsterling_{\xi}\operatorname{Ric}=0$, using this and (13) in the above
equation give (15), and it completes the proof. ∎

An almost contact pseudo-metric structure $(\varphi,\xi,\eta,g)$ is called
normal if $[\varphi,\varphi]+2d\eta\otimes\xi=0$. A normal contact pseudo-
metric manifold is a Sasakian pseudo-metric manifold. A Sasakian pseudo-metric
manifold is a $K-$contact pseudo-metric manifold, satisfying

$\displaystyle(\nabla_{x}\varphi)Y=g(X,Y)\xi-\epsilon\eta(Y)X,$ (16)
$\displaystyle R(X,Y)\xi=\eta(Y)X-\eta(X)Y,$ (17)

where $X,Y\in\Gamma(M)$.

######  Lemma 2.2.

Let $(M,\varphi,\xi,\eta,g)$ be a Sasakian pseudo-metric manifold then
$Q\varphi=\varphi Q$.

###### Proof.

First, calculating the curvature tensor by (16), we have

$\begin{split}R(X,Y,\varphi Z,W)+R(X,&Y,Z,\varphi W)=\\\ &\epsilon g(Z,\varphi
Y)g(X,W)-\epsilon g(Z,\varphi X)g(Y,W)\\\ &-\epsilon g(X,Z)g(\varphi
Y,W)+\epsilon g(Y,Z)g(\varphi X,W),\end{split}$

where $X,Y,Z,W\in\Gamma(M)$ and $R(X,Y,Z,W)=g(R(X,Y,Z),W)$. Now, let $X,Y,Z,W$
be orthogonal to $\xi$, then using the above equation, we have

$R(X,Y,Z,W)=R(\varphi X,\varphi Y,\varphi Z,\varphi W),$

and this gives

$\operatorname{Ric}(X,\varphi Y)+\operatorname{Ric}(\varphi X,Y)=0,$

where $X,Y$ are orthogonal vector fields. Using the last equation, we obtain
$Q\varphi=\varphi Q$, completing the proof. ∎

A contact pseudo-metric manifold $(M,\varphi,\xi,\eta,g)$ is called an
$\eta-$Einstein manifold if the Ricci curvature is of the form
$\operatorname{Ric}=ag+b\eta\otimes\eta$, where $a,b$ are smooth functions on
the manifold $M$. If the manifold $M$ is a $K-$contact pseudo-metric manifold
with dimension greater than three, then $a,b$ are constants.

Let $(M,\varphi,\xi,\eta,g)$ be a contact pseudo-metric manifold, for any
constant real number $t\neq 0$, is defined a contact pseudo-metric manifold
$(M,\tilde{\varphi},\tilde{\xi},\tilde{\eta},\tilde{g})$, where
$\tilde{\eta}=t\eta$, $\tilde{\xi}=\frac{1}{t}\xi$, $\tilde{\varphi}=\varphi$
and $\tilde{g}=tg+\epsilon(t-1)\eta\otimes\eta$. This transition is called a
$\mathcal{D}-$homothetic deformation and it preserves some basic properties
such as being $K-$contact and, in particular being Sasakian. A
$\mathcal{D}-$homothetic deformation of an $\eta-$Einstein $K-$contact pseudo-
metric manifold with $\operatorname{Ric}=ag+b\eta\otimes\eta$ is an
$\eta-$Ricci $K-$contact pseudo-metric manifold such that
$\tilde{\operatorname{Ric}}=(\frac{a-2\epsilon
t+2\epsilon}{t})\tilde{g}+(2n-\tilde{a})\tilde{\eta}\otimes\tilde{\eta}$,
notice that when $a=-2\epsilon$ then the Ricci tensor form is not changed.
Thus, we have the following definition.

######  Definition 1.

An $\eta-$Einstein $K-$contact pseudo-metric manifold with $a=-2\epsilon$ is
said to be $\mathcal{D}-$homothetically fixed.

##  3 $\eta-$Ricci solitons on Sasakian pseudo-metric manifolds

In this section, we have studied $\eta-$Ricci solitons on Sasakian pseudo-
metric manifolds.

######  Theorem 3.1.

Let $(M,\varphi,\xi,\eta,g)$ be a $(2n+1)$-dimensional Sasakian pseudo-metric
manifold. If $(g,V,\lambda,\mu)$ be an $\eta$-Ricci soliton on the manifold
$M$, then $M$ is an $\eta-$Einstein manifold and

$\displaystyle\operatorname{Ric}=(n\epsilon+\frac{\mu\epsilon-\lambda}{2})g+(\frac{n}{2}(\epsilon+1)+\frac{\lambda}{4}(\epsilon+1)+\frac{(\epsilon-3)}{4}\mu)\eta\otimes\eta,$
(18) $\displaystyle
r=\frac{1}{4}\epsilon\left(\lambda-\mu+8n^{2}+(4\mu+6)n\right)+\frac{1}{4}(-\lambda+\mu-4\lambda
n+2n),$ (19)

where $\operatorname{Ric}$ and $r$ are the Ricci tensor and the scalar
curvature of the metric $g$, respectively.

###### Proof.

Using (1) in the following formula [28, p. 23]

$(\mathsterling_{V}\nabla_{x}g-\nabla_{X}\mathsterling_{V}g-\nabla_{[V,X]}g)(Y,Z)=-g((\mathsterling_{V}\nabla)(X,Y),Z)-g((\mathsterling_{V}\nabla)(X,Z),Y),$

where $X,Y$ and $Z$ are arbitrary vector fields, we find

$\begin{split}g((\mathsterling_{V}\nabla)(X,Y),Z)=&(\nabla_{Z}(\operatorname{Ric}+\mu\eta\otimes\eta))(X,Y)\\\
-&(\nabla_{X}(\operatorname{Ric}+\mu\eta\otimes\eta))(Y,Z)\\\
-&(\nabla_{Y}(\operatorname{Ric}+\mu\eta\otimes\eta))(Z,X),\qquad\forall
X,Y,Z\in\Gamma(M).\end{split}$ (20)

Using lemma (2.1) and lemma (2.2) we obtain $\nabla_{\xi}Q=0$ . Substituting
$\xi$ for $Y$ in (20), using the foregoing equation and lemma (2.1) give

$(\mathsterling_{V}\nabla)(X,\xi)=(4n+2\mu)\varphi X-2\epsilon Q\varphi
X,\qquad\forall X\in\Gamma(M).$ (21)

Differentiating (21) along an arbitrary vector field $Y$ and using (16) yield

$\begin{split}(\nabla_{Y}\mathsterling\nabla)(X,\xi)-\epsilon(\mathsterling_{V}\nabla)(X,\varphi
Y)&=2\mu g(X,Y)\xi-(4n+2\mu)\epsilon\eta(X)Y\\\
&-2\epsilon(\nabla_{Y}Q)(\varphi X)+2\eta(X)QY,\qquad\forall
X,Y\in\Gamma(M).\end{split}$ (22)

Using (22) in the following commutative formula [28]

$(\mathsterling_{V}R)(X,Y)Z=(\nabla_{X}\mathsterling_{V}\nabla)(Y,Z)-(\nabla_{Y}\mathsterling_{V}\nabla)(X,Z),$
(23)

where $X,Y,Z$ are arbitrary vector fields, we find:

$\begin{split}(\mathsterling_{V}R)(X,Y)&=2\epsilon(\nabla_{Y}Q)\varphi
X-2\epsilon(\nabla_{X}Q)\varphi Y\\\ &+2\eta(Y)QX-2\eta(X)QY\\\
&+(4n+2\mu)\epsilon\eta(X)Y-(4n+2\mu)\epsilon\eta(Y)X\\\
&+\epsilon(\mathsterling_{V}\nabla)(Y,\varphi
X)-\epsilon(\mathsterling_{V}\nabla)(X,\varphi Y),\qquad\forall
X,Y,Z\in\Gamma(M).\end{split}$ (24)

Substituting $\xi$ for $Y$ in (24) and using (21), we obtain

$(\mathsterling_{V}R)(X,\xi)\xi=4QX-4\epsilon(2n+\mu)X+4\mu\epsilon\eta(X)\xi,\qquad\forall
X\in\Gamma(M).$ (25)

Using (1), we have

$(\mathsterling_{V}g)(X,\xi)+(4n+2\lambda\epsilon+2\mu)\eta(X)=0,\qquad\forall
X\in\Gamma(M),$

and this equation yields

$\displaystyle\epsilon(\mathsterling_{V}\eta)(X)-g(X,\mathsterling_{V}\xi)+2(2n+\lambda\epsilon+\mu)\eta(X)=0,$
(26)
$\displaystyle\eta(\mathsterling_{V}\xi)=(2n\epsilon+\mu\epsilon+\lambda),$
(27)

where $X$ is an arbitrary vector field. Next Lie-differentiating the formula
$R(X,\xi)\xi=X-\eta(X)\xi$ along the vector field $V$ and using (26), (27) and
(17), we have

$QX=\frac{(\epsilon-1)}{4}((\mathsterling_{V}\eta)X)\xi+(n\epsilon+\frac{\mu\epsilon-\lambda}{2})X+(n+\frac{\lambda\epsilon+(1-2\epsilon)\mu}{2})\eta(X)\xi,$
(28)

where $X\in\Gamma(M)$. Now using the foregoing equation and symmetry of the
Ricci tensor, we deduce

$\frac{\epsilon(\epsilon-1)}{4}(\mathsterling_{V}\eta)X\eta(Y)=\frac{\epsilon(\epsilon-1)}{4}(\mathsterling_{V}\eta)Y\eta(X)\qquad\forall
X,Y\in\Gamma(X).$

Using the above equation and (27), we find (18), and in turn it yields (19),
completing the proof. ∎

Theorem 3.1 imposes strong condition on the potential vector field of an
$\eta$-Ricci soliton on a Sasakian pseudo-metric manifold. We need the
following lemma to further study.

######  Lemma 3.2.

Let $(M,\varphi,\xi,\eta,g)$ be a $(2n+1)$-dimensional contact pseudo-metric
manifold. If $\operatorname{Ric}=ag+b\eta\otimes\eta$, where
$a,b\in\mathbf{R}$, then

$\operatorname{Ric}^{ij}\operatorname{Ric}_{ij}+\lambda r+\mu(\epsilon
a+b)=0,$ (29)

where $r$ is the scalar curvature of the metric $g$.

###### Proof.

Using (1) in the following formula [28]

$\mathsterling_{V}\Gamma_{ij}^{h}=\frac{1}{2}g^{ht}\left(\nabla_{j}(\mathsterling_{V}g_{it})+\nabla_{i}(\mathsterling_{V}g_{jt})-\nabla_{t}(\mathsterling_{V}g_{ij})\right),$

where $\Gamma_{ij}^{h}$ are the Christoffel symbols of the metric $g$, we
deduce

$\mathsterling_{V}\Gamma_{ij}^{h}=\nabla^{h}(\operatorname{Ric}_{ij}+\mu\eta_{i}\eta_{j})-\nabla_{i}(\operatorname{Ric}^{h}_{j}+\mu\eta_{j}\eta^{h})-\nabla_{j}(\operatorname{Ric}^{h}_{i}+\mu\eta_{i}\eta^{h}).$

Next using the above equation in the following equation [28]

$\mathsterling_{V}R_{kji}^{h}=\nabla_{k}(\mathsterling_{V}\Gamma_{ij}^{h})-\nabla_{j}(\mathsterling_{V}\Gamma_{ki}^{h}).$

We obtain

$\begin{split}\mathsterling_{V}R^{h}_{kji}&=\nabla_{k}\nabla^{h}(\operatorname{Ric}_{ij}+\mu\eta_{i}\eta_{j})-\nabla_{k}\nabla_{i}(\operatorname{Ric}^{h}_{j}+\mu\eta_{j}\eta^{h})\\\
&-\nabla_{k}\nabla_{j}(\operatorname{Ric}^{h}_{i}+\mu\eta_{i}\eta^{h})-\nabla_{j}\nabla^{h}(\operatorname{Ric}_{ki}+\mu\eta_{k}\eta_{i})\\\
&+\nabla_{j}\nabla_{k}(\operatorname{Ric}_{i}^{h}+\mu\eta_{i}\eta^{h})+\nabla_{j}\nabla_{i}(\operatorname{Ric}_{k}^{h}+\mu\eta_{k}\eta^{h}).\end{split}$

The foregoing equation and the lemma’s assumption yield

$\mathsterling_{V}\operatorname{Ric}_{ji}=\nabla_{h}\nabla^{h}(\operatorname{Ric}_{ij}+\mu\eta_{i}\eta_{j})-\nabla_{h}\nabla_{i}(\operatorname{Ric}^{h}_{i}+\mu\eta_{j}\eta^{h})-\nabla_{h}\nabla_{j}(\operatorname{Ric}_{i}^{h}+\mu\eta_{i}\eta^{h}).$

Eq.(1) gives $\mathsterling_{V}g^{ij}=2\operatorname{Ric}^{ij}+2\lambda
g^{ij}+2\mu\eta^{i}\eta^{j}$, using this, (1) and the above equation we obtain
(29), and it completes the proof. ∎

######  Theorem 3.3.

Let $(M,\varphi,\xi,\eta,g)$ be a Sasakian pseudo-metric manifold and let
$(g,V,\lambda,\mu)$ be an $\eta-$Ricci soliton on $M$.

  1. (a)

If $\xi$ is a timelike vector field then, $V$ is a Killing vector field.

  2. (b)

If $\xi$ is a spacelike vector field and $V$ is not a Killing vector field,
then $M$ is $\mathcal{D}-$homothetically fixed and
$\mathsterling_{V}\varphi=0$.

###### Proof.

In the case of (a), using (12), we find $\lambda-\mu=2n$, this, (18) and (1)
yield $V$ is Killing.

In the case of (b), using lemma (3.2), we obtain
$(-\lambda+\mu+2n+4)(\lambda+\mu+2n)=0$. According to the theorem’s assumption
$V$ is not a Killing vector field, so $\lambda-\mu=2n+4$, using this in (18),
we deduce

$\operatorname{Ric}=-2g+2(n+1)\eta\otimes\eta,$ (30)

so $M$ is $\mathcal{D}-$homothetically fixed. Using the foregoing equation and
(20), we obtain

$(\mathsterling_{V}\nabla)(Y,Z)=2(2n+2+\mu)(\eta(Z)\varphi Y+\eta(Y)\varphi
Z),\qquad\forall Y,Z\in\Gamma(M).$

Differentiating the above equation along an arbitrary vector field $X$, using
(23), and contracting at $X$, we have

$(\mathsterling_{V}\operatorname{Ric})(Y,Z)=2(2n+2+\mu)(2g(Y,Z)-(4n+2)\eta(Y)\eta(Z)),$
(31)

where $Y,Z$ are arbitrary vector fields.

Next using (30) in (1), we find

$(\mathsterling_{V}g)(Y,Z)=-(2n+\lambda+\mu)(g+\eta\otimes\eta)(Y,Z),\qquad\forall
Y,Z\in\Gamma(M).$ (32)

Lie-differentiating (30) along the vector field $V$ gives us

$\begin{split}(\mathsterling_{V}\operatorname{Ric})(Y,Z)=&2(2n+\lambda+\mu)\\{g(Y,Z)+\eta(Y)\eta(Z)\\}+\\\
&2(n+1)\\{\eta(Z)(\mathsterling_{V}\eta)Y+\eta(Y)(\mathsterling_{V}\eta)Z\\},\qquad\forall
Z,Y\in\Gamma(M).\end{split}$ (33)

Substituting $\xi$ for $Y$ in (33) and (31), using (27), we obtain

$(\mathsterling_{V}\eta)Y=-2(2+2n+\mu)\eta(Y),\qquad\forall Y\in\Gamma(M).$
(34)

Operating the above equation by $d$ and noticing the fact that $d$ commutes
with Lie-derivative we deduce

$(\mathsterling_{V}d\eta)(X,Y)=-2(2+2n+\mu)g(X,\varphi Y),\qquad\forall
X,Y\in\Gamma(M).$

Lie-differentiating (5) along the vector field $V$ and using the above
equation, yield $\mathsterling_{V}\varphi=0$, and it completes the proof. ∎

######  Example 1.

Consider $\mathbf{R}^{3}$ with the standard coordinate system $(x,y,z)$. Let
$\xi=2\frac{\partial}{\partial z}$, $\eta=\frac{1}{2}(-ydx+dz)$,
$\varphi(\frac{\partial}{\partial x})=-\frac{\partial}{\partial y}$,
$\varphi(\frac{\partial}{\partial y})=\frac{\partial}{\partial
x}+y\frac{\partial}{\partial z}$ and $\varphi(\frac{\partial}{\partial z})=0$.
If $g=\epsilon\eta\otimes\eta+\frac{1}{4}(dx^{2}+dy^{2})$, then
$(M,\varphi,\xi,\eta,g)$ is a Sasakian pseudo-metric manifold. By direct
calculation, we have $\operatorname{Ric}=-2\epsilon g+4\eta\otimes\eta$. Now,
let $V$ be a vector field defined by

$V=((2-6\epsilon+(\epsilon-1)\lambda+(1-2\epsilon)\mu)x\frac{\partial}{\partial
x}+(2\epsilon-\lambda)y\frac{\partial}{\partial
y}-(2+\epsilon\lambda+\mu)z\frac{\partial}{\partial z}.$

If $\xi$ be a spacelike vector field and $\lambda-\mu=6$ then
$(g,V,\lambda,\mu)$ is an $\eta-$Ricci soliton on $M$,
$\mathsterling_{V}\varphi=0$ and $V$ is not a Killing vector field. But if
$\xi$ is a timelike vector field then $(g,V,\lambda,\mu)$ is an $\eta-$Ricci
soliton on $M$ iff $V$ is a Killing vector field, and this condition is
satisfied if $\lambda=-2$ and $\mu=-4$.

######  proposition 3.4.

Let $(M,\varphi,\xi,\eta,g)$ be a $K$-contact pseudo-metric manifold. If
$(g,V,\lambda,\mu)$ is a gradient $\eta$-Ricci soliton on $M$ then $M$ is an
$\eta-$Einstein manifold and $\operatorname{Ric}=-\lambda
g-\mu\eta\otimes\eta$, where $-\epsilon\lambda-\mu=2n.$

###### Proof.

First (2) gives

$\nabla_{X}Df+QX+\lambda X+\epsilon\mu\eta(X)\xi=0,\qquad\forall
X\in\Gamma(M).$ (35)

Calculating $R(X,Y)Df$ by the above equation, we deduce

$\begin{split}R(X,Y)Df&=\epsilon\mu(\nabla_{Y}\eta)X\xi+\epsilon\mu\eta(X)\nabla_{Y}\xi+(\nabla_{Y}Q)X\\\
&-\epsilon\mu(\nabla_{X}\eta)Y\xi-\epsilon\mu\eta(Y)\nabla_{X}\xi-(\nabla_{X}Q)Y,\qquad\forall
X,Y\in\Gamma(M).\end{split}$ (36)

Substituting $\xi$ for $Y$ in the last equation and using lemma 2.1, we find

$R(X,\xi)Df=(\mu+2n)\varphi X-\epsilon\varphi QX,\qquad X\in\Gamma(M).$

Scalar product of the above equation with $\xi$ gives $df=(\xi(f))\eta$,
operating $d$ on this equation, we obtain $d\eta\wedge(\xi(f))+\eta\wedge
d(\xi(f))=0$, taking exterior product of the last equation with $\eta$ and
using $\eta\wedge d\eta\neq 0$, we have $\xi(f)=0$, so $f$ is a constant
function. Next using this consequence in (35), we find
$\operatorname{Ric}=-\lambda g-\mu\eta\otimes\eta$ and this gives
$-\epsilon\lambda-\mu=2n$, completing the proof. ∎

One may ask, what will happen if the potential vector filed of an $\eta-$Ricci
soliton on a contact pseudo-metric manifold $(M,\varphi,\xi,\eta,g)$ is $\xi$,
we have answered this question in the following theorem.

######  Theorem 3.5.

Let $(M,\varphi,\xi,\eta,g)$ be a contact pseudo-metric manifold, and let
$(g,\varphi,\lambda,\mu)$ be an $\eta-$Ricci soliton on the manifold $M$. If
$V$ is colinear with $\xi$ and $Q\varphi=\varphi Q$, then $M$ is an
$\eta-$Einstein $K-$contact pseudo-metric manifold and
$\operatorname{Ric}=-\lambda g-\mu\eta\otimes\eta$, where
$-\epsilon\lambda-\mu=2n.$

###### Proof.

Let $V=f\xi$, where $f$ is a non-zero smooth function on the manifold $M$.
Using this in (1), we have

$\epsilon X(f)\eta(Y)+\epsilon Y(F)\eta(X)-2fg(\varphi
hX,Y)+2\operatorname{Ric}(X,Y)+2\lambda g(X,Y)+2\mu\eta(X)\eta(Y)=0,$ (37)

for $X,Y\in\Gamma(M)$. Substituting $\xi$ for $Y$ in (37), we deduce

$\epsilon Df+2Q\xi+(\xi(f)+2\lambda+2\epsilon\mu)\xi=0.$ (38)

By assumption $Q\varphi=\varphi Q$, so $\varphi Q\xi=0$, using this and (11),
we have $Q\xi=\epsilon(2n-trh^{2})\xi$. Substituting this consequence in (38),
we find

$\epsilon Df+(2\epsilon(2n-trh^{2})+\xi(f)+2\lambda+2\mu\epsilon)\xi=0.$ (39)

Next, substituting $\xi$ for $X,Y$ in (37), we obtain

$2n-trh^{2}=-\epsilon(\xi(f))-\lambda\epsilon-\mu.$

The above equation and (39) give $Df=\epsilon(\xi(f))\xi$, differentiating
this equation along an arbitrary vector field $X$ and using (10), we find

$g(\nabla_{X}(Df),Y)=X(\xi(f))\eta(Y)-\epsilon\xi(f)\\{g(\epsilon\varphi
X,Y)+g(\varphi hX,Y)\\},\quad\forall X,Y\in\Gamma(M).$

Using the above equation, (5) and the known formula
$g(\nabla_{X}(Df),Y)=g(\nabla_{Y}(Df),X)$, where $X,Y\in\Gamma(M)$, we deduce

$X(\xi(f))\eta(Y)-Y(\xi(f))\eta(X)=-2\xi(f)d\eta(X,Y),\qquad\forall
X,Y\in\Gamma(M).$

Considering $X,Y$ as arbitrary orthogonal vector fields to $\xi$ in the above
equation and noticing that $d\eta\neq 0$, we deduce $X(f)=0$, so $f$ is a
constant function on the manifold $M$. Using this consequence in (37) gives

$-f\varphi hX+QX+\lambda X+\epsilon\mu\eta(X)\xi=0,\qquad\forall
X\in\Gamma(M).$ (40)

Substituting $\varphi X$ for $X$ in the above equation, we find

$-f\varphi h\varphi X+QX+\lambda X=0,\qquad\forall X\in\Gamma(M).$ (41)

Operating $\varphi$ on (40) and using $\varphi h=-h\varphi$, we have

$f\varphi h\varphi X+QX+\lambda X=0,\qquad\forall X\in\Gamma(M).$ (42)

Using the above equation, (41), (3) and $Q\xi=(-\lambda-\mu\epsilon)\xi$, we
obtain:

$\operatorname{Ric}=-\mu\eta\otimes\eta-\lambda g.$

Using the above equation in (1) gives $\mathsterling_{\xi}g=0$, so $M$ is a
$K-$contact pseudo-metric manifold and $-\epsilon\lambda-\mu=2n$, completing
the proof. ∎

##  4 $\eta-$Ricci solitons on $(\kappa,\mu)-$contact pseudo-metric manifolds

Studying nullity conditions on manifolds is one of the interesting topics in
differential geometry, specially in the context of contact pseudo-metric
manifolds. In [15], Ghaffarzadeh and second author introduced the notion of a
$(\kappa,\mu)-$contact pseudo-metric manifold. According to them a contact
pseudo-metric manifold $(M,\varphi,\xi,\eta)$ is called a
$(\kappa,\mu)-$contact pseudo-metric manifold if it satisfies

$R(X,Y)\xi=\epsilon\kappa(\eta(Y)X-\eta(X)Y)+\epsilon\mu(\eta(Y)hX-\eta(X)hY),$
(43)

where $R$ is the Riemannian curvature tensor of $M$, $\kappa,\mu$ are constant
real numbers, and $X,Y$ are arbitrary vector fields. For a
$(\kappa,\mu)-contact$ pseudo-metric manifold we have the following formulas
[15]

$\displaystyle h^{2}=(\epsilon\kappa-1)\varphi^{2},$ (44) $\displaystyle
Q\xi=2n\kappa\xi,$ (45) $\displaystyle(\nabla_{\xi}h)=-\epsilon\mu\varphi h,$
(46)

furthermore if $\epsilon\kappa<1$ then we have [15]

$\displaystyle\begin{split}QX&=\epsilon[2(n-1)-n\mu]X+(2(n-1)+\mu)hX\\\
&+[2(1-n)\epsilon+2n\kappa+n\epsilon\mu]\eta(X)\xi,\end{split}$ (47)
$\displaystyle r=2n(\kappa-2\epsilon)+2n^{2}\epsilon(2-\mu),$ (48)

where $X$ and $r$ are, an arbitrary vector field and the scalar curvature of
the manifold, respectively.

######  Lemma 4.1.

Let $(M,\varphi,\xi,\eta,g)$ be a $(\kappa,\mu)-$contact pseudo-metric
manifold, and let $\epsilon\kappa<1$. If $(g,V,\lambda,\tau)$ is a gradient
$\eta-$Ricci soliton on the manifold $M$ then

$\epsilon\kappa(-2+\mu)=n\mu+\mu+\tau.$ (49)

###### Proof.

Differentiating (45) along an arbitrary vector field $X$ and using (10), we
deduce

$(\nabla_{X}Q)\xi=Q(\epsilon\varphi+\varphi
h)X-2n\kappa(\epsilon\varphi+\varphi h)X,\qquad\forall X\in\Gamma(M).$ (50)

Taking scalar product of (36) and $\xi$, and using (50), we have

$\begin{split}g(R(X,Y)Df,\xi)&=\epsilon g((Q\varphi+\varphi Q)Y,X)+g((Q\varphi
h+h\varphi Q)Y,X)\\\ &+(-4n\kappa\epsilon-2\tau)g(\varphi Y,X),\quad\forall
X,Y\in\Gamma(M).\end{split}$ (51)

Substituting $\varphi X$ for $X$ and $\varphi Y$ for $Y$ in (43) give
$R(\varphi X,\varphi Y)\xi=0$, using this, (3) and the above equation, we
obtain

$\epsilon(\varphi Q+Q\varphi)X-(\varphi
Qh+hQ\varphi)X+(-4n\kappa\epsilon-2\tau)\varphi X=0,$ (52)

where $X$ is an arbitrary vector field. Now, substituting $\varphi X$ for $X$
in (47), we have

$Q\varphi X=\epsilon[2(n-1)-n\mu]\varphi X+(2(n-1)+\mu)h\varphi
X,\qquad\forall X\in\Gamma(M).$ (53)

Next, operating $\varphi$ on (47), we obtain

$\varphi Qx=\epsilon[2(n-1)-n\mu]\varphi X+(2(n-1)+\mu)\varphi
hX,\qquad\forall X\in\Gamma(M).$ (54)

Substituting $hX$ for $X$ in (54), using (44) and (3) give

$\varphi QhX=\epsilon[2(n-1)-n\mu]\varphi
hX-(\epsilon\kappa-1)(2(n-1)+\mu)\varphi X,\qquad X\in\Gamma(M).$ (55)

Operating $h$ on (53), using (44) and (3), we have

$hQ\varphi X=\epsilon[2(n-1)-n\mu]h\varphi
X-(\epsilon\kappa-1)(2(n-1)+\mu)\varphi X,\qquad X\in\Gamma(M).$ (56)

Using the last four equations in (52) and $h\varphi=-\varphi h$ give (49),
completing the proof. ∎

######  Theorem 4.2.

Let $(M,\varphi,\xi,\eta,g)$ be a $(\kappa,\mu)-$contact pseudo-metric
manifold and let $\epsilon\kappa<1$. If $(g,V,\lambda,\tau)$ is a gradient
$\eta-$Ricci soliton on $M$, then $\mu=0,\tau=-2\epsilon\kappa$, or
$\operatorname{Ric}=-\lambda g-\tau\eta\otimes\eta$ and
$\mu=2-2n,\tau=2n(-\frac{1}{n}+n-\epsilon\kappa)$.

###### Proof.

First, substituting $\xi$ for $X$ in (51), using (43) and (45), we have

$\kappa(\xi(f)\xi-\epsilon Df)-\epsilon\mu hDf=0.$

Differentiating the above equation along vector field $\xi$ and using (46), we
have

$\kappa\xi(\xi(f))\xi+\epsilon\kappa(2n\kappa+\lambda+\tau\epsilon)\xi+(\epsilon\mu)^{2}\varphi
hDf=0.$

Now, operating $\varphi$ on the last equation we find $\mu^{2}hDf=0$, taking
$h$ from this and using (44), we obtain

$\mu^{2}(\epsilon\kappa-1)(-Df+\eta(Df)\xi)=0.$

Examining the above equation we have either, i)$\mu=0$ or ii)$\mu\neq 0$.

In the case i), using (49), we obtain $\tau=-2\epsilon\kappa$. In the case
ii), we have $Df=\eta(Df)\xi$, differentiating this along arbitrary vector
field $X$ and using (10), we have

$g(\nabla_{X}Df,Y)=X(\xi(f))\eta(Y)-\xi(f)g(\varphi
X,Y)-\epsilon\xi(f)f(\varphi hX,Y),$

where $X,Y$ are arbitrary vector fields. Using the above equation and
$g(\nabla_{X}Df,Y)=g(\nabla_{Y}Df,X)$, we find

$X(\xi(f))\eta(Y)-Y(\xi(f))\eta(X)+2\xi(f)d\eta(X,Y)=0,\qquad\forall
X,Y\in\Gamma(M).$

Substituting $\varphi X$ for $X$ and $\varphi Y$ for $Y$, and noticing the
fact that $d\eta\neq 0$, we have $\xi(f)=0$. So $f$ is a constant function and
$\operatorname{Ric}=-\lambda g-\tau\eta\otimes\eta$. Using this gives
$r=(2n+1)(-\lambda)-\epsilon\tau$, comparing the last consequence and (48), we
have

$n\mu=-2+2n-2n\kappa\epsilon-\tau.$

Now using the above equation and (49) we obtain, $\mu=2-2n$ and
$\tau=2n(-\frac{1}{n}+n-\epsilon\kappa)$, completing the proof. ∎

######  Corollary 1.

Let $(M,\varphi,\xi,\eta,g)$ be a $(\kappa,\mu)-$contact pseudo-metric
manifold and let $\epsilon\kappa<1$. If $(g,V,\lambda,0)$ is a gradient
$\eta-$Ricci soliton (in fact a gradient Ricci soliton) on $M$, then
$R(X,Y)\xi=0$, where $X,Y$ are arbitrary vector fields.

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# Twisting attosecond pulse trains by amplitude-polarization IR pulses

Enrique G. Neyra<EMAIL_ADDRESS>Centro de Investigaciones Ópticas
(CICBA-CONICET-UNLP), Cno. Parque Centenario y 506, P.O. Box 3, 1897 Gonnet,
Argentina  Fabián Videla  Centro de Investigaciones Ópticas (CICBA-CONICET-
UNLP), Cno. Parque Centenario y 506, P.O. Box 3, 1897 Gonnet, Argentina
Departamento de Ciencias Básicas, Facultad de Ingeniería UNLP, 1 y 47 La
Plata,Argentina  Demian A. Biasetti  Centro de Investigaciones Ópticas (CICBA-
CONICET-UNLP), Cno. Parque Centenario y 506, P.O. Box 3, 1897 Gonnet,
Argentina  Marcelo F. Ciappina<EMAIL_ADDRESS>Department of
Physics, Guangdong Technion - Israel Institute of Technology, 241 Daxue Road,
Shantou, Guangdong, China, 515063  Technion – Israel Institute of Technology,
Haifa, 32000, Israel  Guangdong Provincial Key Laboratory of Materials and
Technologies for Energy Conversion, Guangdong Technion - Israel Institute of
Technology, 241 Daxue Road, Shantou, Guangdong, China, 515063  Lorena Rebón
<EMAIL_ADDRESS>Departamento de Física, FCE, Universidad Nacional de
La Plata, C.C. 67, 1900 La Plata, Argentina  Instituto de Física de La Plata,
UNLP - CONICET, Argentina

###### Abstract

Natively, atomic and molecular processes develop in a sub-femtosecond time
scale. In order to, for instance, track and capture the electron motion in
that scale we need suitable ‘probes’. Attosecond pulses configure the most
appropriate tools for such a purpose. These ultrashort bursts of light are
generated when a strong laser field interacts with matter and high-order
harmonics of the driving source are produced. In this work, we propose a way
to twist attosecond pulse trains. In our scheme, each of the attosecond pulses
in the train has a well-defined linear polarization, but with a different
polarization angle between them. To achieve this goal, we consider an infrared
pulse with a particular polarization state, called amplitude polarization.
This kind of pulse was experimentally synthesized in previous works. Our
twisted attosecond pulse train is then obtained by nonlinear driving an atomic
system with that laser source, through the high-order harmonics generation
phenomenon. We reach a high degree of control in the polarization of the
ultrashort coherent XUV-generated radiation. Through quantum mechanical
simulations, supplemented with signal processing tools, we are able to dissect
the underlying physics of the generation process. We are confident these
polarized-sculpted XUV sources will play an instrumental role in future pump-
probe-based experiments.

Classical electrodynamics and quantum mechanics are the fundamental building
blocks for the description of many natural phenomena. By measuring the
wavelength and speed of light, electromagnetism provides us with tools to
extract the velocity of the light field oscillations. Likewise, quantum
mechanics links the rapidity of electronic motion with the energy distribution
of the populated quantum states. By adequately tuning the light sources, these
states can be accessed by photon absorption and emission. The native scale of
both the light oscillations and electron dynamics falls within the attosecond
range. These elementary processes comprise the constitutional steps of any
change in the physical, chemical, and biological properties of materials and
soft matter. The capability of capturing and manipulating them in real-time is
therefore relevant for the development of novel materials and technologies, as
well as the understanding of fundamental atomic and molecular phenomena
initiated by light fields [1, 2].

Since the first measurement of an attosecond pulse train (APT) [3], attosecond
science has grown enormously, from the obtaining of an isolate attosecond
pulse (IAP) [4] to the generation of optical attosecond pulses [5], producing
a great deal of applications based on these sources [6, 7, 8]. As is well
known, APTs are obtained from the high-order harmonics generation (HHG)
phenomenon. Here, a high-intensity infrared (IR) laser pulse interacts with a
gaseous system producing, at every half laser cycle, a burst of extreme
ultraviolet (XUV) radiation, i.e. a train of ultrashort XUV pulses is
generated. The underlying physics behind the production of APT is rooted in
the so-called three-step model, namely (i) tunneling ionization, (ii)
classical motion, and (iii) recombination. The energy gained by the electron
in its journey through the laser continuum is converted into a high-energy
photon, upon recombination with the parent ion. The strong field approximation
(SFA) can be used to model the HHG process, and, consequently, the generation
of APT [9]. Full-blown quantum mechanical models, based on the numerical
solution of the time-dependent Schrödinger equation (TDSE) can be utilized as
well, although the transparent link between the electron dynamics and their
associated fundamental processes is lost.

The ability to obtain IAPs is key to exploring the electron dynamics in its
natural time scale [4]. Although experimentally it is significantly more
demanding to generate an IAP than an APT, different techniques, namely
amplitude gating, lighthouse, ionization gating, and distinct polarization
gating setups [10, 11], can be used to isolate a single attosecond pulse. On
the other hand, manipulating the spatio-temporal properties or polarization
state of laser pulses, allows other degrees of freedom to be added to the
laser-matter interaction, beyond the frequency and amplitude of the radiation.
To achieve this goal, there exists a great variety of pulse-shaping techniques
[12, 13, 14] which require the use of optical elements and devices such as
wave-plates, spatial light modulators, acousto-optic modulators,
interferometer systems, among others, that can work in the visible and IR
region of the electromagnetic spectrum. However, to have control of the XUV
coherent radiation as in the visible-IR region is highly challenging because,
in general, there are no simple physical devices designed for that wavelength
range. Therefore, the control of the different properties of the coherent XUV
radiation generated via HHG should come from the manipulation of the IR
driving pulse.

Several schemes and sources have been used so far, that allow a high degree of
control in the produced XUV radiation, e.g. attosecond light vortices [15,
16], structured light beams carrying optical angular momentum (OAM) [17],
controlled OAM sources [18, 19], chirality in nonlinear optics [20] and others
[21, 22]. In particular, by using bichromatic and counter-rotating circularly
polarized pulses, it is possible to generate high-order harmonics with
circular polarization, in a gaseous medium [23, 24, 25, 26]. In the temporal
domain, such harmonics originate an APT where the polarization state of each
attosecond pulse depends on the Lissajous curves that describe the bichromatic
driving field, where these curves are shaped like a flower whose petals are
distributed equidistantly on a circumference. Moreover, the temporal structure
of these attosecond pulses has a period that depends on the frequency ratio in
the bichromatic field [27, 28, 25, 29, 30, 31]. Such ratio also gives the
number of petals in the Lissajous curves and the angle between them. This
possibility of manipulating the polarization state of each attosecond pulse in
the APT allows for a more sophisticated level of development and understanding
of ultrafast magnetism [32] in the XUV region, both in molecules [33] and in
solid systems [34]. In addition, it enables the exploration of chiral systems
[35] and the tomographic reconstruction of circularly polarized high-harmonics
fields [36].

In this contribution, we study the HHG in a gaseous target driven by two
single-color orthogonal polarized pulses, which are time delayed from each
other. We model the nonlinear laser-matter electron dynamics through the
numerical solution of the three-dimensional time-dependent Schrödinger
equation (3D–TDSE) in the single-active-electron approximation. We further
analyze the properties in the polarization of the generated APT as well as its
temporal anatomy by invoking signal processing and classical tools. We will
show a procedure to twist the APT by producing a train in which each of the
attosecond pulses has a well-defined linear polarization, but with a different
polarization angle between them. Due to the characteristics of our synthesized
polarization pulse, this technique, in the temporal domain, is similar to the
bichromatic counter-rotating circularly polarized pulses, but with the main
advantage of using a simpler experimental setup, i.e. only a single-color
driving pulse allows controlling the polarization angle between the attosecond
pulses in the train.

##  I Results

###  I.1 Amplitude-polarized pulses

When a linearly polarized laser pulse with an electric field $\bm{E}(t)$,
carrier frequency $\omega_{0}$, field envelope $f(t)$ and global phase $\phi$,
incident upon a birefringent medium with its polarization direction to a given
angle with respect to the optical axis, the field components $E_{x}(t)$ and
$E_{y}(t)$ travel with different velocities in such a medium. Thus, a temporal
delay $\tau$ is generated between them. In particular, for a polarization
direction at an angle of $45^{\circ}$, this field can be mathematically
described as
$\bm{E}(t)=E_{x}(t)\;\check{e}_{x}+E_{y}(t)\;\check{e}_{y}=E_{0}\;f(t)\;\mathrm{cos}(\omega_{0}t+\phi_{x})\;\check{e}_{x}+E_{0}\;f(t-\tau)\;\mathrm{cos}(\omega_{0}t+\phi_{y})\;\check{e}_{y}$,
where $E_{0}$ is the laser electric field peak amplitude, and the phases
$\phi_{x}$ and $\phi_{y}$ take into account the global phase of the field
$\phi$. In general, when the phase difference
$\Delta\phi=\phi_{y}-\phi_{x}=-\omega_{0}\tau\neq n\pi\;(n\in\mathbf{Z})$, it
is said that the pulse has circular ($\Delta\phi=\pm\;\pi/2$) or elliptical
(any other value of $\Delta\phi$) polarization, with ellipticity $\epsilon=1$
($0<\epsilon<1$) for circular (elliptical) polarization. It should be noted
that in the considered case ($\Delta\phi\neq n\pi$), the pulse has a time-
dependent ellipticity $\epsilon(t)$, since the field envelope has a temporal
delay $\tau$. One question immediately arises: How fast is it the variation of
$\epsilon(t)$? The answer can be formulated as follows: given that
$\epsilon(t)$ depends on the bandwidth of the pulse $\Delta\omega$, the
variation will be fast or slow depending on whether
$\Delta\omega\approx\omega_{0}$ or $\Delta\omega\ll\omega_{0}$, respectively.

On the other hand, when the phase difference is $\Delta\phi=n\pi$, the
mathematical description of the field results in
$\bm{E}(t)=E_{0}\;f(t)\;\mathrm{cos}(\omega_{0}t+\phi)\;\check{e}_{x}+E_{0}\;f(t-\tau)\;\mathrm{cos}(\omega_{0}t+n\pi+\phi)\;\check{e}_{y}=E_{0}\;(f(t)\;\check{e}_{x}+f(t-\tau)\;\check{e}_{y})\;\mathrm{cos}(\omega_{0}t+\phi)$,
with $n$ even, and
$\bm{E}(t)=E_{0}\;(f(t)\;\check{e}_{x}-f(t-\tau)\;\check{e}_{y})\;\mathrm{cos}(\omega_{0}t+\phi)$
if $n$ is odd. It is worth mentioning .that the electric field must be written
in this way because the pulse has a bandwidth $\Delta\omega\neq 0$.
Contrariwise, for a monochromatic field, a temporal translation
$\tau=\frac{n\pi}{\omega_{0}}$ does not change the amplitude ratio between the
two orthogonal polarizations.

Experimentally, the polarization scheme described above was presented and
synthesized in Ref. [37], to control the angular momentum orientation of N2
molecules. Later on, other molecular scenarios were studied [38, 39, 40, 41,
42, 43]. This polarization deserves a different classification from linear,
circular or elliptical and, taking into account its mathematical expression,
we will call it amplitude polarization (AP). However, its denomination is not
unified in the literature, and different authors refer to it in different
ways, namely, unidirectional polarization [39], twisted polarization [38],
polarization-shaped pulse [40], and polarization-skewed [41, 44].

In what follows, we will show how this AP field can be used to twist an APT.
In Fig. 1, it can be seen a schematic representation of the experimental setup
necessary to obtain the AP pulses and, through the HHG in a gaseous system,
the twisted APT. Firstly, the IR–AP pulse is generated by a multiple-order
wave plate (MOWP) and a Berek compensator (BC), which are placed in the path
of a single beam, linearly-polarized at $45^{\circ}$ with respect to the
optical axis of the MOWP [37]. This configuration presents a substantial
advantage over other polarization synthesis schemes for the generation or
manipulation of attosecond pulses, where two-color pulses and a complex
interferometric system, are required [27, 28, 25, 29, 36, 34]. In our
proposal, the previously synthesized IR–AP crosses a supersonic gas jet, and
high-order harmonics are generated, whose polarization is sculpted by the
properties of the IR–AP. The resulting APT, after appropriate filtering, can
then be used to drive an atomic, molecular, or solid sample, and track the
multidimensional electron dynamics with attosecond time resolution.

Figure 1: Schematic representation of the experimental setup The IR–AP pulse
is generated by a multiple-order wave plate (MOWP) and a Berek compensator
(BC). These two optical elements are placed in the path of a single beam. The
input IR pulse is linearly-polarized at $45^{\circ}$ with respect to the
optical axis of the MOWP. The synthesized IR–AP crosses a supersonic atomic
gas jet, and high-order harmonics are generated, whose polarization is
controlled by the properties of the IR–AP. The resulting twisted APT is then
filtered and it can be used to study elementary atomic or molecular processes
in its natural time scale.

###  I.2 Twisted attosecond pulse trains

Our aim here is to understand the underlying physics of the interaction of the
IR–AP pulses with an atomic target. To this end, we will compute the HHG
spectra, for different values of the relevant field parameters, on a
prototypical atomic system.

We can start by defining the AP electric field $\bm{E}(t)$ through the
explicit expressions of its components $E_{x}(t)$ and $E_{y}(t)$. For
instance, assuming a Gaussian field envelope $f(t)$, they are written as:

$\displaystyle E_{x}(t)$ $\displaystyle=$ $\displaystyle
E_{0}\;e^{-2\ln(2)\left(\frac{t+\tau/2}{\Delta
t}\right)^{2}}\;\mathrm{cos}(\omega_{0}(t+\tau/2)+\phi)$ (1a) $\displaystyle
E_{y}(t)$ $\displaystyle=$ $\displaystyle
E_{0}\;e^{-2\ln(2)\left(\frac{t-\tau/2}{\Delta
t}\right)^{2}}\;\mathrm{cos}(\omega_{0}(t-\tau/2)+\phi),$ (1b)

where $\Delta t$ is the temporal full width at half maximum (FWHM). For
simplicity, the temporal delay $\tau$ is expressed symmetrically around $t=0$.

Based on the expressions in Eqs. (1a) and (1b), our numerical calculations
were performed for a value $E_{0}=0.053$ a.u., corresponding to a laser
intensity $I_{0}=10^{14}$ W/cm2, and a wavelength $\lambda=800$ nm
($\omega_{0}=0.057$ a.u.). We have analyzed the interaction between the IR–AP
pulses both in the multi- and few-cycle regimes. For the first case, we
considered pulses with an FWHM of $\Delta t=5$ opt. cycles (1 opt. cycle
$=\frac{2\pi}{\omega_{0}}\approx 2.7$ fs). For the second one, we have set the
FWHM to $\Delta t=2$ opt. cycles. By means of the 3D–TDSE in the single active
electron (SAE) approximation (see Refs. [45, 46] for details), we computed the
$x$ and $y$ components of the HHG spectrum, driven by $\bm{E}(t)$. As a
prototypical atomic target, we consider hydrogen (H), with an ionization
potential $I_{p}=0.5$ a.u. Any other atomic target can be adequately modeled
by tuning the parameters of the associated model potential. The temporal delay
$\tau$ is chosen to be $\tau=\Delta t$. For this choice, the peak strength of
the synthesized AP field, $\left|\bm{E}(t)\right|_{\max}$, remains almost
constant and equal to $E_{0}$, in the central region of the pulse (see
Supplementary Information).

In Fig. 2 we show the results obtained by the 3D-TDSE for an AP pulse with a
temporal FWHM of $\Delta t=5$ opt. cycles and a global phase $\phi=0$. Figures
2(a) and 2(b) show the projection of the harmonic radiation in the $x$ and $y$
axis, respectively (see Supplementary Information). Meanwhile, panels (c) and
(d) depict the wavelet analysis of the above-cited spectra and, superimposed,
the respective electric field components $E_{x}(t)$ (white solid line) and
$E_{y}(t)$ (red solid line).

Figure 2: High-order harmonic generation driven by amplitude-polarization (AP)
multi-cycle pulses. Panels (a) and (b): HHG spectra generated by the field
components $E_{x}(t)$ and $E_{y}(t)$, respectively. The FWHM of the AP pulse
is $\Delta t=5$ opt. cycles and the global phase is $\phi=0$. The red regions
in the spectra represent the portion selected to obtain the twisted APTs (see
text for details). Panels (c) and (d): wavelet analysis obtained from the
dipole accelerations $a_{x}(t)$ and $a_{y}(t)$, respectively. The $x$ ($y$)
component of the AP pulse is superimposed with white (red) solid lines. The
black dashed line at the 20th harmonic defines the lower limit of the selected
region.

The core of our results is shown in Fig. 3, where the temporal structure of
the twisted APT can be seen. This temporal representation was obtained after
spectrally filtering the HHG of Figs. 2(a) and 2(b) above the 20th-harmonic.
Then, only the regions of the spectrum colored in red in Figs. 2(a) and 2(b)
are depicted, which in turn correspond to the region above the dashed black
line in Figs. 2(c) and 2(d). This spectral filtering allows us to select the
so-called cut-off region, where the spectral phase of the harmonics is almost
constant and the respective attosecond pulses can be considered Fourier-
limited. On the contrary, it is well-known that in the plateau region of the
HHG spectrum, due to the different recombination times of different electron
trajectories, i.e. different emission times of the radiation (see the wavelet
analysis in Figs. 2(c) and 2(d)), the generation of the attosecond pulses has
an intrinsically spectral phase, the so-called atto-chirp. In general, this
phase can be experimentally compensated by a metallic foil [47].

Figure 3: Temporal anatomy of the twisted APT: multi-cycle pulses regime. (a)
Projection of the twisted APT in the $x$–$y$ plane (solid line) and the field
components $E_{x}(t)$ and $E_{y}(t)$ of the AP pulse (red and blue dashed
lines, respectively). The results that are shown correspond to those of Fig.
2, in the selected region of the spectra. (b) A 3D–representation of the
twisted APT (blue solid line) and the laser electric field of the AP pulse
(red solid line). (c) Lissajous figures of the laser electric field of the AP
pulse (red solid line) and the twisted APT (blue solid line). The shadow
region depicts the angular spacing $\alpha$, between the polarization
direction of two adjacent attosecond pulses. (d) Same as panel (c) but for a
driving AP field with a global phase $\phi=\pi/2$ (only the first quadrant is
plotted). The laser electric field of the AP pulse with a global phase
$\phi=0$ is also shown (red dashed line). The shadow region depicts the
angular spacing $\beta$, between the polarization direction of the two AP
pulses, with global phases $\phi=0$ and $\phi=\pi/2$.

Let us now aim to dissect the twisted temporal anatomy of the APT. For this
purpose we have plotted, in Fig. 3(a), the components of the field $\bm{E}(t)$
(dashed lines) and the twisted APT (solid lines). The $x$-components
($y$-components) are shown in red (blue) color. From there, it can be seen
that the attosecond pulses are generated every half cycle of the AP pulse and,
what is most important, the relative amplitude between the $x$ and $y$
components of the attosecond pulses in the train, changes from one pulse to
the next. This indicates that the polarization between two adjacent attosecond
pulses is different. Furthermore, the full temporal 3D–representation of the
twisted APT can be seen in Fig. 3(b) (blue solid line), where we have also
included the laser electric field of the AP pulse (red solid line). From this
figure, the temporal evolution of each attosecond pulse polarization in the
train and how it rotates is clearly observed by following the temporal
evolution of the AP pulse.

For a better visualization of the polarization features of the individual
attosecond pulses, in Fig. 3(c) we show the Lissajous curves of both, the
laser electric field of the AP pulse (red solid line) and the electric field
of the twisted APT (blue solid line). Two key observations can be extracted
from this figure: First, each attosecond pulse has almost perfect linear
polarization. Second, the polarization direction of each attosecond pulse
follows the orientation of the “petals” that describe the polarization of the
AP electric field. It is possible to obtain the difference in the polarization
direction between two adjacent attosecond pulses, $\alpha$ (see the shadowed
region), which in this case is $\approx 7.5^{\circ}$.

Finally, from Fig. 3(d), we can analyze how the previous results vary by
changing the global phase of the driving AP pulse by $\pi/2$. Considering that
the polarization direction of the attosecond pulses is given by the position
of the petals, for a clear comparison, we show the Lissajous curves of the
electric fields only in the first quadrant. Here, an AP pulse with global
phase $\phi=0$ ($\phi=\pi/2$) is represented by a dashed red (solid black)
line, while the twisted APT is depicted by a blue solid line. The change in
the polarization direction between these AP fields, with different global
phases, is indicated as $\beta$ (see the shadowed region). In this case
$\beta\approx 3.9^{\circ}$, i.e $\beta\approx\alpha/2$. Thus, when the global
phase $\phi$ changes in $\pi/2$, the APT rotates, globally, at an angle equal
to $\beta$.

All the analysis presented above was made in the multi-cycle regime. In order
to complete the picture, in the following we examine the dynamics for a few-
cycle pulse. In Fig. 4 we show our results for an AP pulse with an FWHM of
$\Delta t=2$ opt. cycles. For this FWHM, the angular spacing between two
adjacent petals in the AP pulse is significantly larger than in the previous
multi-cycle case. In order to make a one-to-one comparison, in Figs. 4(a) and
4(b) we show the wavelet analysis coming from the associated HHG spectra in
this few-cycle instance. The $x$ and $y$ components of the AP electric field
$\bm{E}(t)$ are superimposed in white and red solid lines, respectively.
Likewise, in Fig. 4(c) we plot both the electric field components of the AP
field (dashed line) and the APT (solid line). Here, the $x$-components
($y$-components) are shown in red (blue) color. As in the previous case, the
relative amplitude between the components of the attosecond pulses changes
from one pulse to the next. From the Lissajous curve (Fig. 4(d)) a slightly
less ellipticity of the attosecond pulses is observed in relation to that
observed in Fig. 3(d), and a larger angle $\alpha$ between the polarization
direction of two adjacent pulses ($\alpha\approx 20^{\circ}$). The full
temporal evolution of the twisted APT is then shown in Fig. 4(e). As in the
multi-cycle regime, the HHG spectra were spectrally filtered, and we depicted
here only from the 20th-harmonic onwards (see the dashed black line in Figs.
4(a) and 4(b)).

Figure 4:  Temporal anatomy of the twisted APT: few-cycle pulses version. Same
that Figs. 2(c) and 2(d) (in this figure, panels (a) and (b), respectively),
and Figs. 3(a), 3(b) and 3(c) (in this figure, panels (c), (d) and (e),
respectively), but obtained from an AP pulse with a temporal FWHM of $\Delta
t=2$ opt. cycles.

Further analysis of the value of $\alpha$ as a function of the temporal FWHM
and the time delay $\tau$, can be seen in SM. As a summary, we can mention
that $\alpha$ increases almost linearly whit $\tau$, which gives us a clear
idea of how to manipulate the change in the polarization direction between two
adjacent attosecond pulses from the synthesis of the driving IR field.
Furthermore, the calculation of the recollision angles of the ionized
electrons [45], as well as a classical analysis of the trajectories of the
electrons in the continuum, are included in the SM.

##  II Discussion

In this work, we have introduced a straightforward procedure to generate an
APT in which each of the attosecond pulses has a well-defined linear
polarization, but different between them. This appealing feature of the APT is
achieved through the manipulation of the polarization state of the driving IR
field. Here, by introducing a variable time delay between two identical copies
of an IR pulse, we obtain an AP pulse. Classically speaking, the generation of
this twisted APT is rooted in the small deviation of the trajectories of the
electrons in the continuum by the AP pulse, due to the temporal change in its
polarization state. This makes the electrons recombine at different angles,
where the “short” trajectories slightly deviate in relation to the “long”
ones. Furthermore, this deviation is more pronounced with few-cycle driving
pulses, which allows for reaching larger rotation angles. Thus, this small
deviation of the electrons’ trajectories in relation to the linear
polarization pulse, allows us to extend the current results by employing
different conventional sources to obtain an APT, such as high-energy laser
sources, at a high repetition rate, or with different wavelengths. Moreover,
our scheme is quite versatile and robust, as it can be implemented in both the
multi- and few-cycle regimes.

The concrete possibility to manipulate the polarization angle between the
attosecond pulses in an APT would bring attosecond-based spectroscopy
techniques to another, more advanced, level [48, 49, 50, 51, 52]. For
instance, it would be possible to generate photoelectrons within a well-
defined angular range or design complex pump-probe techniques. Just to cite a
few examples, we could think of several pump-probe schemes, namely an IR
pulse/a twisted APT, an AP pulse/a twisted APT, or a twisted APT/twisted APT.
The latter would be an XUV pump-XUV probe scheme, but when each of the pulses
is a train of twisted isolated attosecond pulses. In addition, because we can
manipulate the polarization angle between two adjacent pulses in the APT, one
can imagine sophisticated attosecond-based spectroscopy techniques using a
single beam. Beyond photoelectron spectroscopy, this twisted APT can be useful
to (i) study or drive highly anisotropic systems, for instance, systems where
there are preferential directions, such as molecular systems, low-dimensional
crystalline structures, etc. [7] and (ii) characterize multidimensional laser
fields in the time domain utilizing the well-known streaking technique, mainly
used for linearly polarized fields [53], amongst other.

Likewise, our approach presents clear advantages compared to other isolated
attosecond pulse generation techniques (polarization gating, lighthouse,
amplitude gating, or ionization gating [47]). First, the possibility to work
with multi-cycle pulses (in most of the isolated attosecond pulses generation
techniques few/single-cycle pulses are needed). Second, the synthesis of the
AP pulses is experimentally straightforward: only a single color pulse without
the need for an involved interferometric system is necessary, i.e, we work in
a single beam path. Third, if the spectral phase of the harmonics is
compensated, for example with a metallic foil, it would be possible to use a
large spectral bandwidth, generating ultrashort attosecond pulses.

Finally, let us emphasize the instrumental role of the global phase $\phi$ as
an additional ‘knob’ in our proposed scheme. For instance, we have proved that
a change of the global phase in $\pi/2$ of the driving AP pulse echoed by a
global rotation of the twisted APT. This implies that it is necessary to
stabilize the global phase (carrier-envelope phase) of the AP pulse for pump-
probe experiments or if we aim for coherent control in some processes. On the
contrary, this global rotation of the twisted APT can be useful to detect the
change in the global phase for the AP pulse, both in the few-cycle and multi-
cycle regimes.

##  III Methods

To find the quantum dipole acceleration
$\bm{a}(t)=a_{x}(t)\;\check{e}_{x}+a_{y}(t)\;\check{e}_{y}$, we have
numerically solved the 3D–TDSE [45, 46], in the dipole approximation, by
expanding the active electron wavefunction in spherical harmonics
$Y^{m}_{l}(\theta,\phi)$:

$\Psi(\bm{r},t)=\sum\frac{R^{m}_{l}(\bm{r},t)}{r}Y^{m}_{l}(\theta,\phi),$ (2)

where $\theta$ and $\phi$ are the polar and azimuthal angles. Through the
Ehrenfest’s theorem, the $x$ and $y$ components of the acceleration operator
$\hat{\bm{a}}(t)$ can be found as

$\displaystyle\hat{a}_{x}$ $\displaystyle=$
$\displaystyle-[\hat{H},[\hat{H},\hat{x}]]\;=\;\left[\hat{H},\frac{\partial}{\partial
r}\right]\;\mathrm{cos(\theta)},$ (3a) $\displaystyle\hat{a}_{y}$
$\displaystyle=$
$\displaystyle-[\hat{H},[\hat{H},\hat{y}]]\;=\;\left[\hat{H},\frac{\partial}{\partial
r}\right]\;\mathrm{sin(\theta)cos(\phi)},$ (3b)

where $\hat{H}$ is the full Hamiltonian. Thus, the components of $\bm{a}(t)$
can be obtained as: $a_{x}(t)=\expectationvalue{\hat{a}_{x}}{\Psi(\bm{r},t)}$
and $a_{y}(t)=\expectationvalue{\hat{a}_{y}}{\Psi(\bm{r},t)}$. The projections
in the $x-y$ plane of the harmonics spectra amplitudes
$|\tilde{a}_{x}(\omega)|$ and $|\tilde{a}_{y}(\omega)|$ (Figs. 2(a) and 2(b)),
and the respective spectral phases, $\gamma_{x}(\omega)$ and
$\gamma_{y}(\omega)$, are then calculated by the Fourier transforms

$\displaystyle\tilde{a}_{x}(\omega)$ $\displaystyle=$
$\displaystyle|\tilde{a}_{x}(\omega)|e^{i\gamma_{x}(\omega)}\;=\;\frac{1}{(t_{f}-t_{i})\;\omega^{2}}\int_{t_{i}}^{t_{f}}e^{-i\omega
t}a_{x}(t)\;dt$ (4a) $\displaystyle\tilde{a}_{y}(\omega)$ $\displaystyle=$
$\displaystyle|\tilde{a}_{y}(\omega)|e^{i\gamma_{y}(\omega)}\;=\;\frac{1}{(t_{f}-t_{i})\;\omega^{2}}\int_{t_{i}}^{t_{f}}e^{-i\omega
t}a_{y}(t)\;dt,$ (4b)

where the times $t_{i}$ and $t_{f}$ define the integration window, both to
solve the 3D–TDSE and to evaluate these integrals. For the first case, of the
two analyzed in this work, where the FWHM of the pulse is $\Delta t=5$ opt.
cycles, the numerical calculations were performed for $t_{i}=-2757$ a.u. and
$t_{i}=2757$ a.u., with a time step of 1 a.u. In the second case, where the
FWHM of the pulse is $\Delta t=2$ opt. cycles, the integration times are
$t_{i}=-1103$ a.u. and $t_{i}=1103$ a.u., with a time step of $1$ a.u.

Once we obtain $\bm{a}(t)$, a wavelet analysis can be performed by computing
the Gabor transform, for both the $x$ and $y$ components, as follows:

$G_{x,y}(\omega,t)=\left|\int
a_{x,y}(t^{\prime})\;\frac{\mathrm{exp}[-(t-t^{\prime})^{2}/2\sigma^{2}]}{\sigma\sqrt{2\pi}}\;\mathrm{exp}(i\omega
t^{\prime})\;dt^{\prime}\right|^{2}.$ (5)

A value of $\sigma=1/(3\omega_{0})$ allows achieving an adequate balance
between the time and frequency resolutions. $G_{x,y}(\omega,t)$ then gives us
access to study the time dynamics behind the HHG process and compare it with
classical simulations.

The $x$–$y$ components of the electric field of the different APT,
$E_{XUV,x}(t)$ and $E_{XUV,y}(t)$, are calculated by filtering the harmonic
spectrum, and starting from the 20th ($\omega_{i}=20$) harmonic order:

$\displaystyle E_{XUV,x}(t)$ $\displaystyle=$
$\displaystyle\int_{\omega_{i}=20\omega_{0}}^{\infty}e^{i\omega
t}\tilde{a}_{x}(\omega)\;d\omega,$ (6a) $\displaystyle E_{XUV,y}(t)$
$\displaystyle=$
$\displaystyle\int_{\omega_{i}=20\omega_{0}}^{\infty}e^{i\omega
t}\tilde{a}_{x}(\omega)\;d\omega.$ (6b)

## Acknowledgements

M. F. C. acknowledges financial support from the Guangdong Province Science
and Technology Major Project (Future functional materials under extreme
conditions - 2021B0301030005). E. G. N. acknowledges to Consejo Nacional de
Investigaciones Científicas y Técnicas (CONICET). F. A. V. acknowledges to
Comisión de Investigaciones Científicas de la Pcia. de Buenos Aires.

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