#  Quasinormal modes of gravitational perturbation around some well-known
regular black holes

Chen Wu1111Electronic address<EMAIL_ADDRESS>1\. Shanghai Institute of
Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China

###### Abstract

In this paper, the quasinormal modes of gravitational perturbation around some
well-known regular black holes were evaluated by using the WKB approximation
as well as the asymptotic iteration method. Through numerical calculation, we
make a detailed analysis of the gravitational QNM frequencies by varying the
characteristic parameters of the gravitational perturbation and the spacetime
charge parameters of the regular black holes. It is found that the imaginary
part of quasinormal modes as a function of the charge parameter has different
monotonic behaviors for different black hole spacetimes. Moreover, the
asymptotic expressions of gravitational QNMs for $l\gg 1$ are obtained by
using the eikonal limit method. We demonstrate that the gravitational
perturbation is stable in all these spacetimes.
Key words: gravitational perturbation, regular black hole spacetime,
quasinormal modes

###### pacs:

04.70.Bw, 04.30.-w

##  I Introduction

The investigations concerning the interaction of black holes with various
fields around give us the possibility to get some information about the
physics of black holes. One of these information can be obtained from
quasinormal modes (QNMs) which are characteristic of the background black hole
spacetimes black hole . The concept of QNMs is first formulated by
Vishveshwara in calculations of the scattering of gravitational waves by a
black hole Vishveshwara , which present complex frequencies whose real part
represents the actual frequency of the oscillation and the imaginary part
represents the damping. The survey of field perturbation in black hole
spacetimes motivated the extensive numerical and analytical study of QNMs some
papers . In addition, the properties of QNMs have been studied in the context
of the AdS/CFT correspondence asd/cft  and loop quantum gravity loop . Some
reviews where a lot of references to the recent research of QNMs can be found
in Refs. review .

Among several types of field perturbation, gravitational perturbation is
considered to be the most important one. The reason is that gravitational QNMs
are important in directly identifying black holes and their gravitational
radiation. Many theoretical physicists believe that the gravitational QNMs is
a unique fingerprint in searching the existence of a black hole. Recently,
astrophysical interests in QNMs originated from their relevance in
gravitational wave analysis. On September 14th, 2015, two advanced detectors
of the Laser Interferometer Gravitational-wave Observatory (LIGO) made the
first direct measurement of gravitational waves GW . The Advanced LIGO
detectors observed a transient gravitational-wave signal determined to be the
coalescence of two black holes, launching the era of gravitational wave
astronomy. The issue of black hole gravitational stability under perturbations
was first addressed by Regge and Wheeler wheeler  in the fifties of last
century. They classified gravitational perturbations into two types: odd
parity and even parity by means of getting rid of the angular dependence of
the perturbation variables through a tensorial generalization of the spherical
harmonics, thus the calculation of gravitational perturbation was greatly
simplified. The Regge-Wheeler formalism was later extended to the case of
static black holes in four dimensions G QNMs 4D-1 ; G QNMs 4D-2 ; G QNMs 4D-3
and higher dimensions G QNMs HD-1 , and even to the case of rotating black
holes G QNMs Rotating-1 ; G QNMs Rotating-2 . One can find a complete
description of black hole perturbation theory in the book by Chandrasekhar G
QNMs book .

On the other hand, the problem of understanding how to avoid singularities in
black hole spacetimes is important in general relativity. In 1968, a “regular”
black hole without a singularity was constructed by Bardeen Bardeen . This
regular black hole spacetime lacked a consistent physical interpretation until
Ayón-Beato and his coworkers ABG2000  obtained this black hole solution by
describing it as the gravitational field of nonlinear magnetic monopole with a
mass $M$ and a charge $q$ in 2000.

The Bardeen solution has motivated deeper works about singularity avoidance
may be realized generally. Several other researchers paid attention to
theories of gravity coupled to nonlinear electrodynamics, and proposed other
solutions in different contexts. Some solutions that are relevant to this work
are analyzed in Refs. Hayward ; Bronnikov ; Dymnikova ; ABG ; new1  and Refs.
Bardeen ; ABG2000 .

Recently, there are several interesting works concerning the regular black
hole. Eiroa and Sendra have investigated gravitational lensing of the regular
black hole spacetime eiroa1 . In Ref. f(r) , exact solutions of spherically
symmetric spacetimes are proposed in $f(R)$ modified theories of gravity
coupled to nonlinear electrodynamics. The dynamical stability of black hole
solutions in self-gravitating nonlinear electrodynamics with respect to linear
gravitational fluctuation has been studied in stability regularBH . Fernando
and Correa have studied QNMs spectrum of the scalar field of the regular black
hole for various values of the perturbation parameters Fernando . They have
also used the unstable null geodesics of the black hole to compute the scalar
QNMs in the eikonal limit. Further research about the QNMs of neutral and
charged scalar field perturbations on the regular black hole spacetime in a
variety of models was carried out by Flachi and Lemos Flachi . Massless and
massive Dirac QNMs were studied in the regular black hole spacetime by using
the WKB approach in Ref. Dirac .

In this paper, we concentrate on the behavior of the gravitational
perturbation in the regular black hole spacetimes mentioned above. The
wavelike perturbation equation, with an effective potential, can be solved by
several methods, such as integration of the wavelike equations, the monodromy
method, fit and interpolation approaches, the continued fraction method, the
Mashhoon method, the WKB approximation method, the asymptotic iteration method
and so on QNM method . We calculate the gravitational QNMs by using the 3th
order WKB method as well as the asymptotic iteration method.

The rest of the paper is organized as follows. In section 2, we gives brief
description of some well-known regular black hole spacetimes. The perturbative
equation of the gravitational perturbation in given backgrounds is reduced to
the Schrödinger-like wave equation in section 3. The next section is devoted
to the numerical calculations of the gravitational QNMs in given spacetimes by
using the 3th order WKB approximation and the asymptotic iteration method. The
eikonal limit for the gravitational frequencies is also presented. The
conclusions are given in last section.

##  II The basic equations and numerical results

In this section, we will first give a brief introduction to the regular black
holes. In order to obtain the regular solutions, the typical action to include
the nonlinear electrodynamic term is,

$S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}\left(R-\mathscr{L}(F)\right),$ (1)

where $g$ is the determinant of the black hole metric, $G$ is the
gravitational constant, $R$ is the scalar curvature, and $\mathscr{L}(F)$
represents the Lagrangian of the nonlinear electrodynamics with
$F=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, where
$F_{\mu\nu}=\bigtriangledown_{\mu}A_{\nu}-\bigtriangledown_{\nu}A_{\mu}$ is
the electromagnetic field strength.

The general line element for spherically symmetric regular black hole
solutions can be described as

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}~{},$
(2)

where $(t,r,\theta,\phi)$ are the usual space-time spherical coordinates, and
specific choices of the lapse function $f(r)$ distinguish between the
different spacetimes.

In Ref. Bardeen , the lapse function $f(r)$ for the Bardeen black hole is
determined by the formula

$f(r)=1-\frac{2Mr^{2}}{(r^{2}+q^{2})^{3/2}}\,,$ (3)

where $q$ and $M$ are the magnetic charge and the mass of the magnetic
monopole. Ayón-Beato and his coworker interpreted this black hole as the
gravitational field of a magnetic monopole arising from nonlinear
electrodynamics ABG2000 . The Lagrangian of the specific nonlinear
electrodynamics is given by
$\mathscr{L}(F)=(3M/|q|^{3})(\sqrt{2q^{2}F}/(1+\sqrt{2q^{2}F}))^{5/2}$. The
spacetime in Eq. 3 has horizons only if $|q|\leq\frac{4}{3\sqrt{3}}$. For
$q=\frac{4}{3\sqrt{3}}$, there are degenerate horizons. For
$q>\frac{4}{3\sqrt{3}}$, there are no horizons.

In 2006, Hayward found new regular black hole spacetime that has center
flatness and is quiet similar to the physical insight of the Bardeen one
Hayward . The simple regular black hole implies a specific matter energy-
momentum tensor that is de Sitter at the core and vanishes at large distances
$r\rightarrow\infty$. The function $f(r)$ for the Hayward black hole also
takes a simple form

$f(r)=1-\frac{2Mr^{2}}{r^{3}+2\alpha^{2}}\,$ (4)

with $\alpha=$ Const. As like the Bardeen black hole, the Eq. (4) can also
have zero, one, or two horizons depending on the relative values of $M$ and
$\alpha$.

Through introducing the Lagrangian for nonlinear electrodynamics to first
order, the regular black hole is also constructed originally by Bronnikov
Bronnikov . The lapse function for this black hole is given as

$f=1-{2M\over r}\left(1-\mbox{tanh}{r_{0}\over r}\right),$ (5)

where the parameter $r_{0}$ is a length scale related to the electric charge.

Then Dymnikova Dymnikova  put forward an exact, regular spherically symmetric,
charged black hole solution by using the idea proposed by Bronnikov Bronnikov
. This solution is constructed from a nonlinear electrodynamic theory with a
Hamiltonian-like function (for more details, please see Ref. Dymnikova ). The
lapse function for Dymnikova’s solution is given as

$f(r)=1-\frac{4M}{\pi r}\left(\mbox{tan}^{-1}{r\over r_{0}}-{rr_{0}\over
r^{2}+r_{0}^{2}}\right)~{}.$ (6)

The parameter $r_{0}$ in the solution (6) is a length scale defined as
$r_{0}=\pi q^{2}/(8M)$, where $M$ and $q$ is the total mass and the charge.

Another famous regular black hole is the one presented by Ayón-Beato and
García in Ref. ABG . To find this regular black hole solution, they took the
nonlinear electric field as a source of charge for the solution of Maxwell
field equations. Its lapse function $f(r)$ is given as

$f=1-{2Mr^{2}\over(r^{2}+q^{2})^{3/2}}+{q^{2}r^{2}\over(r^{2}+q^{2})^{2}}~{},$
(7)

where $M$ and $q$ are the total mass and charge. This solution can be obtained
from a nonlinear electrodynamics with Lagrangian density
${\mathscr{L}}(F)={X^{2}\over-2q^{2}}{1-8X-3X^{2}\over\left(1-X\right)^{4}}-{3M\over
2q^{3}}{X^{5/2}\left(3-2X\right)\over\left(1-X\right)^{7/2}}~{},$ where
$X=\sqrt{-2q^{2}F}$.

Recently Balart and Vagenas new1  built charged regular black holes in the
framework of Einstein$-$nonlinear electrodynamics theory. They constructed the
general lapse function for mass distribution functions that are inspired by
continuous probability distributions. In this paper, we consider two examples
of black hole solutions employing their methodology. One metric function is of
the form

$f=1-\frac{2M}{r}e^{-\frac{q^{2}}{2Mr}}~{}.$ (8)

The other metric function is written as

$f=1-\frac{2M}{r}\frac{2}{\text{exp}({\frac{q^{2}}{Mr}})+1}~{}.$ (9)

In two metric functions $M$ and $q$ are associated with total mass and charge,
respectively.

All the black hole solutions used in this paper are summarized in Table 1. The
mass of black holes is normalized to 1. It is noteworthy that in all the
mentioned black holes the lapse function $f(r)$ can have zero, one, or two
horizons depending on the value of charge parameters. In Table 1, we list the
extreme charge parameters for which the inner horizon and outer horizon
coincide. To illustrate the behavior of the lapse function more clearly, we
show the lapse function as a function of $r$ for three values of the charge
parameter in Fig. 1. Noting that the plots of all lapse functions for regular
black holes under consideration show similar behaviors, we take the plot of
the lapse function for the Bardeen regular black hole as an example.

Table 1: Summary of some well-known regular black holes in the paper. For more details, one can refer to the Refs. Bardeen ; ABG2000 ; Hayward ; Bronnikov ; Dymnikova ; ABG ; new1  listed in the second section. In this work, the mass of black holes is normalized to 1. Lapse function | Reference | Extremal condition | Originator
---|---|---|---
$f(r)=1-\frac{2Mr^{2}}{(r^{2}+q^{2})^{3/2}}$ | Bardeen  | $q\approx 0.77$ | Bardeen
$f=1-{2Mr^{2}\over r^{3}+2\alpha^{2}}$ | Hayward  | $\alpha\approx 1.06$ | Hayward
$f=1-{2M\over r}\left(1-\mbox{tanh}{r_{0}\over r}\right)$ | Bronnikov  | $r_{0}\approx 0.55$ | Bronnikov
$f=1-{4M\over\pi r}\left(\mbox{tan}^{-1}{r\over r_{0}}-{rr_{0}\over r^{2}+r_{0}^{2}}\right)$ | Dymnikova  | $r_{0}=0.45$ | Dymnikova
$f=1-{2Mr^{2}\over(r^{2}+q^{2})^{3/2}}+{q^{2}r^{2}\over(r^{2}+q^{2})^{2}}$ | ABG  | $q\approx 0.63$ | Ayón-Beato and García
$f=1-\frac{2M}{r}e^{-q^{2}/{2mr}}$ | new1  | $q\approx 1.21$ | Balart and Vagenas
$f=1-\frac{2M}{r}\frac{2}{e^{q^{2}/{mr}}+1}$ | new1  | $q\approx 1.06$ | Balart and Vagenas

##  III  Perturbation equation for the gravitational field

The investigation of black hole perturbations was first carried out by Regge
and Wheeler wheeler  for the odd parity type of the spherical harmonics and
was extended to the even parity type by Zerilli zerilli .

We indicate the background metric with $g_{\mu\nu}$ and the perturbation in it
with $h_{\mu\nu}$. The perturbation $h_{\mu\nu}$ is very small compared with
$g_{\mu\nu}$. The $R_{\mu\nu}$ can be expressed from $g_{\mu\nu}$, and
$R_{\mu\nu}+\delta R_{\mu\nu}$ from $g_{\mu\nu}+h_{\mu\nu}$. $\delta
R_{\mu\nu}$ can be calculated from the form book

$\delta
R_{\mu\nu}=-\delta\Gamma^{\beta}_{\mu\nu;\>\beta}+\delta\Gamma^{\beta}_{\mu\beta;\>\nu},$
(10)

where

$\delta\Gamma^{\alpha}_{\beta\gamma}=\frac{1}{2}g^{\alpha\nu}(h_{\beta\nu;\>\gamma}+h_{\gamma\nu;\>\beta}-h_{\beta\gamma;\>\nu}).$
(11)

The canonical form for the perturbations in the Regge-Wheeler gauge is given
as wheeler

$h_{\mu\nu}=\begin{array}[]{|cccc|}0&0&0&h_{0}(r)\\\ 0&0&0&h_{1}(r)\\\
0&0&0&0\\\ h_{0}(r)&h_{1}(r)&0&0\end{array}\exp(-i\omega
t)\sin\theta\frac{\partial Y_{L0}}{\partial\theta}$ (12)

Substituting Eq. (12) into Eq. (10), we get

$\frac{1}{f(r)}i\omega
h_{0}(r)-\frac{d}{dr}f(r)h_{1}(r)=0,\;\;\text{from}\;\;\delta R_{23}=0$ (13)
$\frac{1}{f(r)}i\omega\left(i\omega
h_{1}(r)+\frac{d}{dr}h_{0}(r)-\frac{2}{r}h_{0}(r)\right)+\frac{1}{r^{2}}(l(l+1)-2)h_{1}(r)=0,\;\;\text{from}\;\;\delta
R_{13}=0$ (14)

Defining

$\phi=f(r)h_{1}(r)/r,$ (15)

and eliminating the $h_{0}(r)$ we get

$(\frac{d^{2}}{dr_{*}^{2}}+\omega^{2})\Phi(r)=V(r)\Phi(r),$ (16)

where $r_{*}$ is the tortoise coordinate and

$V(r)=f(r)\left(\frac{l(l+1)}{r^{2}}-2\frac{1-f(r)}{r^{2}}-\frac{1}{r}\frac{df(r)}{dr}\right).$
(17)

As mentioned before, the complex $\omega$ values are written as
$\omega=\text{Re}(\omega)+i\text{Im}(\omega)$. The effective potential $V(r)$
of wavelike perturbation equation for the black hole spacetimes is plotted to
display how it changes with the angular harmonic index $l$ in Fig. 2. One can
find from Fig. 2 that the angular harmonic index $l$ increases the height of
the potential barrier governed by the effective potential.

##  IV Numerical methods and numerical results

Now we report the QNM frequencies of the gravitational perturbation in the
regular black hole spacetimes. In order to calculate QNMs, boundary conditions
are imposed for the wavelike perturbation equation. The boundary condition at
the horizon is for the solution to be purely ingoing wave, while the other
boundary condition at spatial infinity is such that the wave has to be purely
outgoing one. Therefore one can write the boundary conditions as

$\Phi(r)\sim e^{-i\omega
r_{*}},\;\;\;\;\text{as}\;\;\;\;r_{*}\rightarrow-\infty,$ (18) $\Phi(r)\sim
e^{i\omega r_{*}},\;\;\;\;\text{as}\;\;\;\;r_{*}\rightarrow+\infty.$ (19)

WKB method. The Schrödinger-like wave equation (16) with the effective
potential (17) containing the lapse function f(r) related to the regular black
holes is not solvable analytically. Many numerical methods are developed to
compute QNMs of various black hole spacetimes in the literature. One of these
standard methods is the WKB approximative method that was applied for the
first time by Schutz and Will WKB . Iyer and his coworkers developed the WKB
method up to third order WKB3  and later, Konoplya developed it up to sixth
order WKB6 . This semianalytic method has been applied extensively in numerous
black hole spacetime cases, which has been proved to be accurate up to around
one percent for the real and the imaginary parts of the quasinormal
frequencies for low-lying modes with $n<l$, where $n$ is the mode number and
$l$ is the angular momentum quantum number. In this paper, the third order WKB
formalism is applied since the 6th order WKB method consumes too much CPU
power for some cases of regular black holes.

The 3th order formalism of the WKB approximation presented in the paper WKB3
has formula

$\frac{\omega^{2}-V_{0}}{\sqrt{-2V_{0}^{{}^{\prime\prime}}}}-L_{2}-L_{3}=n+\frac{1}{2}.$
(20)

Here, $V_{0}$ and $V_{0}^{{}^{\prime\prime}}$ are the maximum potential and
the second derivative of the potential evaluated at the maximum potential, $n$
is the node number, and $L_{n}$ represents the n-th order correction. The
formulae for $L_{2}$ and $L_{3}$ are given in WKB3 .

The eikonal limit. It is well-know that the WKB method works with high
accuracy for large values of the multipole quantum number. For $l\gg 1$,
frequency of QNMs of gravitational perturbation field can be found
analytically by using the first order WKB approach. Since the lapse function
is complicated in any regular black hole metric, a more convenient method is
used to obtain the asymptotic form of frequency of QNMs of gravitational
perturbation field. For $l\gg 1$, the effective potential $V(r)$ at its
maximum has a asymptotic form: $V(r_{Max})\propto l(l+1)$. Then the QNM
frequency takes the form

$\displaystyle\lim_{l\to\infty}\omega\approx
c_{1}(l+\frac{1}{2})-ic_{2}(n+\frac{1}{2})\ .$ (21)

We can readily fix $c_{1}$ and $c_{2}$ by setting $l\gg 1$ in numerical
calculations. In Table. 2, we list different $c_{1}$ and $c_{2}$ for several
regular black hole spacetimes.

Table 2: The constant $c_{1}$ and $c_{2}$ in the eikonal limit for given regular black hole spacetimes. Lapse function | Ref. | Parameter | $c_{1}$ | $c_{2}$
---|---|---|---|---
$f(r)=1-\frac{2Mr^{2}}{(r^{2}+\alpha^{2})^{3/2}}$ | Bardeen  |  $\alpha$ = 0.6 | 0.2066 | 0.1796
$f=1-{2Mr^{2}\over r^{3}+2\alpha^{2}}$ | Hayward  | $\alpha=0.7$ | 0.2009 | 0.1683
$f=1-{2M\over r}\left(1-\mbox{tanh}{r_{0}\over r}\right)$ | Bronnikov  | $r_{0}=0.4$ | 0.2301 | 0.1947
$f=1-{4M\over\pi r}\left(\mbox{tan}^{-1}{r\over r_{0}}-{rr_{0}\over r^{2}+r_{0}^{2}}\right)$ | Dymnikova  | $r_{0}=0.4$ | 0.2491 | 0.1867
$f=1-{2Mr^{2}\over(r^{2}+q^{2})^{3/2}}+{q^{2}r^{2}\over(r^{2}+q^{2})^{2}}$ | ABG  | $q=0.6$ | 0.2267 | 0.1712
$f=1-\frac{2M}{r}e^{-q^{2}/{2mr}}$ | new1  | $q=0.6$ | 0.2053 | 0.1961
$f=1-\frac{2M}{r}\frac{2}{e^{q^{2}/{mr}}+1}$ | new1  | $q=0.6$ | 0.4493 | 0.3917

The asymptotic iteration method. The asymptotic iteration method (AIM) was
first applied to solve the second order differential equations Ciftci . This
new method was then used to obtain the QNM frequencies of field perturbation
in Schwarzschild black hole spacetime AIM2010 .

Let’s consider a second order differential equation of the form

$\chi^{\prime\prime}=\lambda_{0}(x)\chi^{\prime}+s_{0}(x)\chi,$ (22)

where $\lambda_{0}(x)$ and $s_{0}(x)$ are well defined functions and
sufficiently smooth. Differentiating the equation above with respect to $x$
leads to

$\chi^{\prime\prime\prime}=\lambda_{1}(x)\chi^{\prime}+s_{1}(x)\chi,$ (23)

where the new two coefficients are
$\lambda_{1}(x)=\lambda_{0}^{\prime}+s_{0}+\lambda_{0}^{2}$ and
$s_{1}(x)=s_{0}^{\prime}+s_{0}\lambda_{0}.$

Using this process iteratively, we can get the $(n+2)-$th derivative of
$\chi(x)$ with respect to $x$ as

$\chi^{(n+2)}=\lambda_{n}(x)\chi^{\prime}+s_{n}(x)\chi,$ (24)

where the new coefficients $\lambda_{n}(x)$ and $s_{n}(x)$ are associated with
the older ones through the following relation

$\lambda_{n}(x)=\lambda^{\prime}_{n-1}+s_{n-1}+\lambda_{0}\lambda_{n-1},\;\;\;\;s_{n}(x)=s^{\prime}_{n-1}+s_{0}\lambda_{n-1}.$
(25)

For sufficiently large values of $n$, the asymptotic concept of the AIM method
is introduced by AIM2012 ,

$\frac{s_{n}(x)}{\lambda_{n}(x)}=\frac{s_{n-1}(x)}{\lambda_{n-1}(x)}=\text{Constant}.$
(26)

The perturbation frequencies can be obtained from the above “quantization
condition”. However this procedure has a difficulty in that the process of
taking the derivative of $\lambda_{n}(x)$ and $s_{n}(x)$ terms terms of the
previous iteration at each step can consume much time and affect the numerical
precision of calculations. To overcome these drawbacks, $\lambda_{n}(x)$ and
$s_{n}(x)$ are expanded in Taylor series around the point $x^{\prime}$ at
which the AIM method is performed AIM2010 ,

$\displaystyle\lambda_{n}(x^{\prime})=\sum_{i=0}^{\infty}c_{n}^{i}(x-x^{\prime})^{i}$
, $\displaystyle
s_{n}(x^{\prime})=\sum_{i=0}^{\infty}d_{n}^{i}(x-x^{\prime})^{i},$ (27)

where $c_{n}^{i}$ and $d_{n}^{i}$ are the $i$-th Taylor coefficients of
$\lambda_{n}(x^{\prime})$ and $s_{n}(x^{\prime})$, respectively. Substitution
of above equations into Eq. (25) leads to a set of recursion relations for the
Taylor coefficients as

$\displaystyle
c^{i}_{n}=\sum_{k=0}^{i}c_{0}^{k}c_{n-1}^{i-k}+(i+1)c_{n-1}^{i+1}+d^{i}_{n-1}$
, $\displaystyle
d_{n}^{i}=\sum_{k=0}^{i}d_{0}^{k}c_{n-1}^{i-k}+(i+1)d^{i+1}_{n-1}.$ (28)

After applying the recursion relations (28) in Eq. (26), the quantization
condition then can be expressed as

$d_{n}^{0}c_{n-1}^{0}-c_{n}^{0}d_{n-1}^{0}=0,$ (29)

which can be employed to calculate the QNMs of a black hole. Both the accuracy
and efficiency of the AIM method are greatly improved without any derivative
computation AIM2010 .

First, we calculate gravitational QNM frequencies by varying the charge of the
Bardeen regular black hole. We also calculate the gravitational QNM
frequencies for the Reissner-Nordstr$\ddot{\text{o}}$m black hole with the
same charge for comparison as shown in Table 3. Here, the node number $n=0$
and the angular momentum quantum number $l=2$. From Table. 3, it is shown
clearly that the real part $\text{Re}(\omega)$ of the gravitational QNMs
increases when charge parameter $q$ increases for the two black hole
spacetimes. And the imaginary value $\text{Im}(\omega)$ of QNMs decreases for
the Bardeen black hole spacetime with $q$, while for the Reissner-
Nordstr$\ddot{\text{o}}$m black hole spacetime, the imaginary value arrives a
maximum 0.0909 around $q=0.76$.

Table 3: The gravitational QNMs in the Bardeen black hole spacetime as a
function of the
charge parameter. As comparison, we also calculate the QNMs in the Reissner-Nordstr$\ddot{\text{o}}$m black hole spacetime. The numerical results are computed by the AIM method. Here, $n=0$ and $l=2$. q |  $\omega$ (RN black hole) |  $\omega$ (Bardeen black hole)
---|---|---
0.1 | 0.3743 $-$ 0.0890i | 0.3743 $-$ 0.0889i
0.2 | 0.3763 $-$ 0.0892i | 0.3765 $-$ 0.0885i
0.3 | 0.3797 $-$ 0.0894i | 0.3801 $-$ 0.0879i
0.4 | 0.3847 $-$ 0.0898i | 0.3856 $-$ 0.0870i
0.5 | 0.3916 $-$ 0.0902i | 0.3933 $-$ 0.0855i
0.6 | 0.4008 $-$ 0.0906i | 0.4039 $-$ 0.0829i
0.7 | 0.4130 $-$ 0.0909i | 0.4189 $-$ 0.0782i
0.76 | 0.4244 $-$ 0.0909i | 0.4309 $-$ 0.0728i
0.8 | 0.4297 $-$ 0.0908i | **
0.85 | 0.4404 $-$ 0.0903i | **

Results for the quasinormal frequencies for gravitational perturbations in
regular black hole spacetimes under consideration are tabulated, in Table 4
for the solution given in Refs. Bardeen ; ABG2000 , in Table 5 for the
solution given in Ref. Hayward , in Table 6 for the solution given in Ref.
Bronnikov , in Table 7 for the solution given in Ref. Dymnikova , in Table 8
for the solution given in Ref. ABG , in Table 9 for the solution given in Ref.
new1 , and in Table 10 for the solution given in Ref. new1 . The values of the
quasinormal frequencies listed in these tables are computed by the third order
WKB method (without parenthesis) and the AIM method (with parenthesis),
respectively. It has been shown that the linear perturbative gravitational
field are stable around all of the considered regular black holes.

We can apply the sixth order WKB approximation to check the convergence of the
WKB approximation. In Fig. 3, The real and imaginary parts of the QNMs from
gravitational perturbations for the $l=2,n=0$ mode for the model of Bardeen
Bardeen  are presented when higher order terms in the WKB approximation are
included in the computation. It shows that the accuracy of the third order WKB
method is reliable.

The above calculations have shown that increasing of the spacetime charge
parameter implies monotonic increasing of the real part of quasinormal
frequency. However, the imaginary part of quasinormal frequency as a function
of the charge parameter has different monotonic behaviors for different black
hole spacetimes.

For the Bardeen spacetime, the Hayward spacetime and the solution of Ref. ABG
, increasing of the spacetime charge parameter implies monotonic decreasing of
the imaginary part of QNM frequency; For the two solutions of Ref. new1 , the
imaginary part of QNM frequency increases when the charge parameter increases;
However, for the two solutions of Ref. Bronnikov ; Dymnikova , there exists a
maximum of the imaginary part in the charge parameter interval.

Table 4:  QNMs for gravitational perturbations for $q=$ 0.1, 0.3, 0.6 for the model of Refs. Bardeen ; ABG2000 .  $l$ | n | $q=0.1$ | $q=0.3$ | $q=0.6$
---|---|---|---|---
2 | 0 | 0.3738 $-$ 0.0891i | 0.3795 $-$ 0.0881i | 0.4030 $-$ 0.0828i
|  | (0.3743 $-$ 0.0889i) | (0.3801 $-$ 0.0879i) | (0.4039 $-$ 0.0829i)
2 | 1 | 0.3468 $-$ 0.2745i | 0.3539 $-$ 0.2712i | 0.3820 $-$ 0.2536i
|  | (0.3476 $-$ 0.2736i) | (0.3552 $-$ 0.2705i) | (0.3850 $-$ 0.2534i)
3 | 0 | 0.6003 $-$ 0.0926i | 0.6091 $-$ 0.0917i | 0.6455 $-$ 0.0866i
|  | (0.6005 $-$ 0.0926i) | (0.6093 $-$ 0.0917i) | (0.6457 $-$ 0.0866i)
3 | 1 | 0.5835 $-$ 0.2811i | 0.5933 $-$ 0.2782i | 0.6328 $-$ 0.2620i
|  | (0.5838 $-$ 0.0809i) | (0.5936 $-$ 0.2781i) | (0.6335 $-$ 0.2619i)
3 | 2 | 0.5546 $-$ 0.4761i | 0.5659 $-$ 0.4708i | 0.6102 $-$ 0.4421i
|  | (0.5531 $-$ 0.4785i) | (0.5648 $-$ 0.4730i) | (0.6107 $-$ 0.4429i)
Table 5:  QNMs for gravitational perturbations for $\alpha=0.1,0.4,0.7$ for the model of Refs. Hayward .  $l$ | n | $\alpha=0.1$ | $\alpha=0.4$ | $\alpha=0.7$
---|---|---|---|---
2 | 0 | 0.3735 $-$ 0.0891i | 0.3782 $-$ 0.0864i | 0.3903 $-$ 0.0776i
|  | (0.3740 $-$ 0.0888i) | (0.3791 $-$ 0.0864i) | (0.3920 $-$ 0.0781i)
2 | 1 | 0.3464 $-$ 0.2744i | 0.3521 $-$ 0.2659i | 0.3619 $-$ 0.2378i
|  | (0.3472 $-$ 0.2734i) | (0.3549 $-$ 0.2652i) | (0.3696 $-$ 0.2374i)
3 | 0 | 0.5997 $-$ 0.0926i | 0.6072 $-$ 0.0902i | 0.6470 $-$ 0.0820i
|  | (0.5999 $-$ 0.0926i) | (0.6075 $-$ 0.0902i) | (0.6274 $-$ 0.0820i)
3 | 1 | 0.5829 $-$ 0.2810i | 0.5914 $-$ 0.2734i | 0.6108 $-$ 0.2478i
|  | (0.5832 $-$ 0.2808i) | (0.5922 $-$ 0.2733i) | (0.6126 $-$ 0.2477i)
3 | 2 | 0.5539 $-$ 0.4759i | 0.5638 $-$ 0.4626i | 0.5806 $-$ 0.4182i
|  | (0.5524 $-$ 0.4783i) | (0.5638 $-$ 0.4643i) | (0.5834 $-$ 0.4177i)
Table 6:  QNMs for gravitational perturbations for $\alpha=0.1,0.3,0.4$ for the model of Refs. Bronnikov . The parameter $r_{0}$ is a length scale related to the charge.  $l$ | n | $r_{0}=0.1$ | $r_{0}=0.3$ | $r_{0}=0.4$
---|---|---|---|---
2 | 0 | 0.3872 $-$ 0.0902i | 0.4240 $-$ 0.0911i | 0.4499 $-$ 0.0903i
|  | (0.3877 $-$ 0.0900i) | (0.4246 $-$ 0.0911i) | (0.4505 $-$ 0.0904i)
2 | 1 | 0.3611 $-$ 0.2774i | 0.4014 $-$ 0.2794i | 0.4298 $-$ 0.2763i
|  | (0.3620 $-$ 0.2766i) | (0.4027 $-$ 0.2791i) | (0.4316 $-$ 0.2760i)
3 | 0 | 0.6213 $-$ 0.0938i | 0.6791 $-$ 0.0949i | 0.7197 $-$ 0.0942i
|  | (0.6215 $-$ 0.0937i) | (0.6793 $-$ 0.0949i) | (0.7198 $-$ 0.0942i)
3 | 1 | 0.6051 $-$ 0.2844i | 0.6652 $-$ 0.2874i | 0.7073 $-$ 0.2847i
|  | (0.6054 $-$ 0.2843i) | (0.6655 $-$ 0.2873i) | (0.7078 $-$ 0.2846i)
3 | 2 | 0.5772 $-$ 0.4813i | 0.6409 $-$ 0.4855i | 0.6855 $-$ 0.4802i
|  | (0.5758 $-$ 0.4836i) | (0.6400 $-$ 0.4871i) | (0.6852 $-$ 0.4812i)
Table 7:  QNMs for gravitational perturbations for $\alpha=0.1,0.3,0.4$ for the model of Refs. Dymnikova .  $l$ | n | $r_{0}=0.1$ | $r_{0}=0.3$ | $r_{0}=0.4$
---|---|---|---|---
2 | 0 | 0.3914 $-$ 0.0904i | 0.4444 $-$ 0.0908i | 0.4888 $-$ 0.0867i
|  | (0.3919 $-$ 0.0902i) | (0.4451 $-$ 0.0908i) | (0.4894 $-$ 0.0867i)
2 | 1 | 0.3657 $-$ 0.2772i | 0.4239 $-$ 0.2778i | 0.4714 $-$ 0.2638i
|  | (0.3666 $-$ 0.2773i) | (0.4255 $-$ 0.2775i) | (0.4736 $-$ 0.2634i)
3 | 0 | 0.6279 $-$ 0.0940i | 0.7112 $-$ 0.0946i | 0.7806 $-$ 0.0904i
|  | (0.6281 $-$ 0.0940i) | (0.7114 $-$ 0.0946i) | (0.7808 $-$ 0.0904i)
3 | 1 | 0.6120 $-$ 0.2850i | 0.6985 $-$ 0.2861i | 0.7698 $-$ 0.2728i
|  | (0.6123 $-$ 0.2850i) | (0.6989 $-$ 0.2860i) | (0.7704 $-$ 0.2727i)
3 | 2 | 0.5845 $-$ 0.4821i | 0.6762 $-$ 0.4827i | 0.7500 $-$ 0.4587i
|  | (0.5731 $-$ 0.4826i) | (0.6758 $-$ 0.4839i) | (0.7507 $-$ 0.4589i)
Table 8:  QNMs for gravitational perturbations for $q=0.0,0.3,0.6$ for the model of Refs. ABG .  $l$ | n | $q=0.0$ | $q=0.3$ | $q=0.6$
---|---|---|---|---
2 | 0 | 0.3732 $-$ 0.0892i | 0.3860 $-$ 0.0884i | 0.4438 $-$ 0.0788i
|  | (0.3737 $-$ 0.0890i) | (0.3866 $-$ 0.0883i) | (0.4447 $-$ 0.0790i)
2 | 1 | 0.3460 $-$ 0.2749i | 0.3609 $-$ 0.2721i | 0.4240 $-$ 0.2400i
|  | (0.3467 $-$ 0.2739i) | (0.3623 $-$ 0.2714i) | (0.4275 $-$ 0.2396i)
3 | 0 | 0.5993 $-$ 0.0927i | 0.6193 $-$ 0.0921i | 0.7097 $-$ 0.0826i
|  | (0.5994 $-$ 0.0927i) | (0.6195 $-$ 0.0921i) | (0.7099 $-$ 0.0826i)
3 | 1 | 0.5824 $-$ 0.2814i | 0.6038 $-$ 0.2792i | 0.6975 $-$ 0.2492i
|  | (0.5826 $-$ 0.2813i) | (0.6042 $-$ 0.2791i) | (0.6985 $-$ 0.2490i)
3 | 2 | 0.5532 $-$ 0.4767i | 0.5771 $-$ 0.4724i | 0.6749 $-$ 0.4191i
|  | (0.5517 $-$ 0.4791i) | (0.5760 $-$ 0.4745i) | (0.6762 $-$ 0.4189i)
Table 9:  QNMs for gravitational perturbations for $q=0.1,0.3,0.6$ for the first model of Refs. new1 .  $l$ | n | $q=0.1$ | $q=0.3$ | $q=0.6$
---|---|---|---|---
2 | 0 | 0.3738 $-$ 0.0893i | 0.3791 $-$ 0.0897i | 0.3992 $-$ 0.0909i
|  | (0.3743 $-$ 0.0890i) | (0.3796 $-$ 0.0894i) | (0.3997 $-$ 0.0908i)
2 | 1 | 0.3467 $-$ 0.2751i | 0.3524 $-$ 0.2761i | 0.3741 $-$ 0.2796i
|  | (0.3474 $-$ 0.2741i) | (0.3532 $-$ 0.2752i) | (0.3750 $-$ 0.2789i)
3 | 0 | 0.6003 $-$ 0.0928i | 0.6087 $-$ 0.0932i | 0.6402 $-$ 0.0946i
|  | (0.6005 $-$ 0.0928i) | (0.6089 $-$ 0.0932i) | (0.6404 $-$ 0.0946i)
3 | 1 | 0.5834 $-$ 0.2816i | 0.5921 $-$ 0.2828i | 0.6246 $-$ 0.2868i
|  | (0.5837 $-$ 0.2814i) | (0.5923 $-$ 0.2827i) | (0.6249 $-$ 0.2867i)
3 | 2 | 0.5543 $-$ 0.4769i | 0.5634 $-$ 0.4787i | 0.5977 $-$ 0.4852i
|  | (0.5528 $-$ 0.4793i) | (0.5619 $-$ 0.4813i) | (0.5964 $-$ 0.4873i)
Table 10:  QNMs for gravitational perturbations for $q=0.1,0.3,0.6$ for the second model of Refs. new1 .  $l$ | n | $q=0.1$ | $q=0.3$ | $q=0.6$
---|---|---|---|---
2 | 0 | 0.7489 $-$ 0.1786i | 0.7713 $-$ 0.1801i | 0.8773 $-$ 0.1817i
|  | (0.7499 $-$ 0.1781i) | (0.7724 $-$ 0.1797i) | (0.8786 $-$ 0.1817i)
2 | 1 | 0.6948 $-$ 0.5504i | 0.7190 $-$ 0.5544i | 0.8351 $-$ 0.5564i
|  | (0.6962 $-$ 0.5484i) | (0.7206 $-$ 0.5528i) | (0.8382 $-$ 0.5559i)
3 | 0 | 1.2026 $-$ 0.1857i | 1.2379 $-$ 0.1873i | 1.4043 $-$ 0.1893i
|  | (1.2030 $-$ 0.1856i) | (1.2382 $-$ 0.1873i) | (1.4046 $-$ 0.1893i)
3 | 1 | 1.1689 $-$ 0.5634i | 1.2053 $-$ 0.5682i | 1.3781 $-$ 0.5729i
|  | (1.1695 $-$ 0.5632i) | (1.2059 $-$ 0.5680i) | (1.3789 $-$ 0.5728i)
3 | 2 | 1.1108 $-$ 0.9544i | 1.1491 $-$ 0.9618i | 1.3326 $-$ 0.9669i
|  | (1.1078 $-$ 0.9592i) | (1.1463 $-$ 0.9663i) | (1.3315 $-$ 0.9695i)

##  V summary

Although we do not have a complete theory of quantum gravity, regular black
hole solutions were proposed by coupling Einstein gravity to an external form
of matter. Therefore it is interesting to compute QNMs for the regular black
hole spacetimes and see how it is different from the ordinary ones. In this
work, we have studied the quasinormal modes of gravitational perturbation
around some well-known regular black hole by using the WKB approximation and
the asymptotic iteration method. Through numerical calculation, we made a
detailed analysis of the gravitational QNM frequencies by varying the
characteristic parameters of the gravitational perturbation and the spacetime
charge parameters of the regular black holes. Numerical results show that the
imaginary part of quasinormal modes as a function of the charge parameter has
different monotonic behaviors for different black hole spacetimes. The
asymptotic expressions of gravitational QNMs for $l\gg 1$ are computed by
using the eikonal limit method. It is demonstrated that the gravitational
perturbation is stable in all these spacetimes.

###### Acknowledgements.

We thank Prof. Ru-Keng Su for very helpful discussions. We like to thank R.A.
Konoplya for providing the WKB approximation. We also like to thank developers
of the AIM method for their opening codes. This work is supported partially by
the Major State Basic Research Development Program in China (No.
2014CB845402).

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Figure 1: Radial dependence of lapse function $f(r)$ of the Bardeen regular
black hole for different values of the charge parameter.  Figure 2: Variation
of the effective potential $V(r)$ with respect to the radial coordinate $r$
for three values of the angular harmonic index $l$.  Figure 3:  Re($\omega$)
and Im($\omega$) parts of the gravitational QNMs of the Bardeen black hole
with $q=0.1$ as a function of WKB order are given for the $n=0,\;\;l=2$ and
$n=0,\;\;l=2$ modes.


Detecting Phishing Sites Using ChatGPT

Takashi Koide

NTT Security Japan (KK)

Naoki Fukushi

NTT Security Japan (KK)

Hiroki Nakano

NTT Security Japan (KK)

Daiki Chiba

NTT Security Japan (KK)

The rise of large language models (LLMs) has had a significant impact on various domains, including natural language processing and artificial intelligence. While LLMs such as ChatGPT have been extensively researched for tasks such as code generation and text synthesis, their application in detecting malicious web content, particularly phishing sites, has been largely unexplored. To combat the rising tide of automated cyber attacks facilitated by LLMs, it is imperative to automate the detection of malicious web content, which requires approaches that leverage the power of LLMs to analyze and classify phishing sites.

In this paper, we propose a novel method that utilizes ChatGPT to detect phishing sites. Our approach involves leveraging a web crawler to gather information from websites and generate prompts based on this collected data. This approach enables us to detect various phishing sites without the need for fine-tuning machine learning models and identify social engineering techniques from the context of entire websites and URLs. To evaluate the performance of our proposed method, we conducted experiments using a dataset. The experimental results using GPT-4 demonstrated promising performance, with a precision of 98.3% and a recall of 98.4%. Comparative analysis between GPT-3.5 and GPT-4 revealed an enhancement in the latter's capability to reduce false negatives. These findings not only highlight the potential of LLMs in efficiently identifying phishing sites but also have significant implications for enhancing cybersecurity measures and protecting users from the dangers of online fraudulent activities.

#1     #1#1#1     #1     #1     #1     #1#1          #1#1

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# Coherent control of magnetization precession by double-pulse activation of
effective fields from magnetoacoustics and demagnetization

M. Mattern  Institut für Physik & Astronomie, Universität Potsdam, 14476
Potsdam, Germany  F.-C. Weber  Institut für Physik & Astronomie, Universität
Potsdam, 14476 Potsdam, Germany  D. Engel  Max-Born-Institut für Nichtlineare
Optik und Kurzzeitspektroskopie, 12489 Berlin, Germany  C. Korff-Schmising
Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, 12489
Berlin, Germany  M. Bargheer<EMAIL_ADDRESS>Institut für Physik &
Astronomie, Universität Potsdam, 14476 Potsdam, Germany  Helmholtz-Zentrum
Berlin für Materialien und Energie, 12489 Berlin, Germany

###### Abstract

We demonstrate the coherent optical control of magnetization precession in a
thin Ni film by a second excitation pulse which amplifies or attenuates the
precession induced by a first pulse depending on the fluences of the pump-
pulses and the pump-pump delay. This control goes beyond the conventional
strategy, where the same mechanism drives the precession in or out-of phase.
We balance the magneto-acoustic mechanism driven by quasi-static strain and
the shape-anisotropy change triggered by laser-induced demagnetization. These
mechanisms tilt the transient effective magnetic field in opposite directions
in case of negative magneto-elastic coupling ($b_{1}<0$). While the strain
response is linear in the fluence, demagnetization is nonlinear near the Curie
temperature, enabling fluence-based control scenarios.

††preprint: AIP/123-QED

Since the discovery of ultrafast demagnetization in thin Ni films[1], the
field of ultrafast magnetization dynamics has evolved from observing ultrafast
magnetization dynamics [2, 3] to controlling the magnetization on ultrafast
timescales[4, 5, 6, 7, 8].

The fastest ways to coherently change the magnetization state is the
excitation of precessional motion reported for a wide range of ferro- or
ferri-magnetic metals[9, 10, 11], semiconductors[12] and insulators [13, 14].
Depending on the magnetically ordered material, the precession can be driven
by temperature-induced changes of the magnetocrystalline anisotropy [11, 10],
ultrafast demagnetization affecting the shape anisotropy [11, 9, 15, 16] or by
both local quasi-static expansion [17, 15] and propagating strain pulses [12,
18, 19, 20, 16] via material-specific magneto-elastic coupling mediated by
spin-orbit coupling.

In classical coherent control experiments [21, 22, 19, 23, 24] the
magnetization precession is tuned by the pump-pump delay in a double-pulse
excitation scheme, where both excitations induce the same driving mechanism,
which depends linearly on the fluence. Very recently, Shin and co-workers[17,
15] controlled the precession amplitude and phase via the composition of a
NixFe100-x film, which tunes the magneto-elastic coupling $b_{1}$, the Curie-
temperature $T_{\text{C}}$ and the saturation magnetization $M_{\text{sat}}$.
This changes the dominant driving mechanism from quasi-static expansion to
laser-induced demagnetization that counteract in materials with negative
magneto-elastic coupling ($b_{1}<0$). In addition, they showed this
counterbalance to depend on the fluence due to the non-linear increase of the
demagnetization near $T_{\text{C}}$ that becomes the dominant driving
mechanism and switches the precession phase by $\pi$.

In this letter, we combine coherent control of the magnetization precession by
double-pulse excitation with tuning of the dominant driving mechanism via the
fluences of both pump-pulses and the pump-pump delay. The counterbalance of
laser-induced demagnetization and the negative magneto-elastic coupling
($b_{1}<0$) in our $20\,\text{nm}$-thin Ni film strongly depends on the laser-
induced temperature rise and the initial sample temperature since the
demagnetization increases non-linearly near $T_{\text{C}}$. We demonstrate a
transition of the dominant driving mechanism of the second excitation from
magneto-elastic coupling to demagnetization in three scenarios: The
temperature before the second pulse is tuned via the fluence of the first
excitation or the pump-pump delay or, alternatively, the fluence of the second
pulse determines the temperature rise. The change of the dominant driving
mechanism switches the effect of the second pulse from amplifying to
attenuating the precession induced by the first pulse, which opens rich
opportunities to control the precession by double-pulse excitation.

In our all-optical transient polar magneto-optical Kerr experiments (trMOKE)
sketched in Fig. 1(a), a $20\,\text{nm}$ Ni film embedded within a
$2\,\text{nm}$ Pt capping and $3\,\text{nm}$ Pt and Ta buffer layers is
excited by two p-polarized $120\,\text{fs}$-long $800\,\text{nm}$ pump-pulses
from the substrate-side at normal incidence with tunable pump-pump delay
$\Delta t$ and individually adjustable fluences $F_{1}$ and $F_{2}$. We probe
the transient out-of-plane magnetization of the Ni film by detecting the
polarization rotation of a $400\,\text{nm}$ probe pulse that is reflected at
near normal incidence. The external magnetic field of around $350\,\text{mT}$
is applied along $35^{\circ}$ with respect to the surface normal set by a
rotatable permanent magnet[16] and is below the out-of-plane saturation field
$\mu_{0}H_{\text{sat}}=500\,\text{mT}$.

Figure 1(b) displays the fluence-dependent transient out-of-plane
magnetization upon a single excitation that induces demagnetization and drives
a coherent magnetization precession. With increasing fluence, the out-of-plane
magnetization is reduced more and more and we observe a critical slowing down
of the remagnetization[25]. While the amplitude of the demagnetization
increases with increasing fluence, the amplitude of the coherent magnetization
precession increases only for small fluences
($<2.9\,\text{mJ}\,\text{cm}^{-2}$). For further increased fluence the
precession is diminished again, such that the amplitude for $5.0$ and
$0.7\,\text{mJ}\,\text{cm}^{-2}$ become comparable.

This non-linear fluence dependence originates from the opposite effect that
laser-induced expansion and demagnetization have on the orientation of the
effective magnetic field in materials with negative magneto-elastic coupling
[17]. The effective field
$\mu_{0}\vec{H}_{\text{eff}}=-\nabla_{M}F\textsubscript{M}$ is derived from
the Free energy $F_{\text{M}}$ of the macroscopic magnetization and determines
the orientation of the magnetization vector $\vec{M}=M_{\text{sat}}\vec{m}$.
For our thin polycrystalline Ni film grown by magnetron sputtering on a PGO
substrate this Free energy is given by a Zeeman energy originating from the
external magnetic field $\vec{H}_{\text{ext}}$, a shape anisotropy determined
by the thin film geometry and a magneto-elastic contribution induced by the
mean out-of-plane strain $\eta_{\text{Ni}}(t)$[16, 15]:

$\begin{split}F\textsubscript{M}(\vec{m},t)&=-\mu_{0}M_{\text{sat}}\vec{m}(t)\cdot\vec{H}\textsubscript{ext}+\frac{\mu_{0}M_{\text{sat}}^{2}}{2}m_{z}(t)^{2}\\\
\
&{\color[rgb]{1,1,1}=}\,+b_{1}\eta_{\text{Ni}}(t)\,m_{z}(t)^{2}\;.\end{split}$
(1)

The latter two contributions lead to effective fields that are proportional to
the out-of-plane component of the magnetization vector $m_{z}$. According to
Eq. 1, ultrafast demagnetization, i.e. a rapid drop of $m_{z}(t)$, and the
laser-induced strain response of Ni
$\eta_{\text{Ni}}(t)=\eta_{\text{qs}}(t)+\eta_{\text{sp}}(t)$ composed of a
quasi-static expansion $\eta_{\text{qs}}$ and propagating strain pulses
$\eta_{\text{sp}}$ are the driving mechanisms of the magnetization precession.
Since the round-trip time of the strain pulse through the metallic
heterostructure of $\approx 5\,\text{ps}$ is an order of magnitude shorter
than the precession period, the contribution of $\eta_{\text{sp}}$ becomes
negligible[20] and the laser-induced demagnetization and quasi-static
expansion of Ni remain as driving mechanisms. Due to the negligible in-plane
saturation field of $\mu_{0}H_{\text{sat}}=5\,\text{mT}$ representing the
magnetic easy axis, we also neglect any contribution from a magneto-
crystalline anisotropy[16, 15].

These mechanisms induce a change of the effective field
$\Delta\vec{H}_{\text{eff}}$ that adds to the effective field before
excitation $\vec{H}_{\text{eff},0}$, that determines the initial direction of
the magnetization. This tilting of the effective field triggers precessional
motion of $\vec{M}$ according to the Landau-Lifshitz-Gilbert equation[10].
Figure 2(a) sketches the resulting effective field upon laser excitation
$\vec{H}_{\text{eff}}=\vec{H}_{\text{eff},0}+\Delta\vec{H}_{\text{eff}}$ and
the concomitant precession for laser-induced demagnetization (left) and quasi-
static expansion (right). While the demagnetization tilts
$\vec{H}_{\text{eff}}$ out-of-plane by adding a $\Delta\vec{H}_{\text{eff}}$
in the positive $z$ direction, quasi static expansion has the opposite effect.
Therefore, the induced magnetization precession starts in opposite directions
for the individual mechanisms. The corresponding phase shift of $\pi$ results
in a destructive superposition of the two effects. The total effective field
change $\vec{H}_{\text{eff},0}$ depends on the temperature increase $\Delta T$
and the temperature before excitation $T_{\text{init}}$. The quasi-static
strain $\eta_{\text{qs}}$ depends linearly on $\Delta T$ and is insensitive to
$T_{\text{init}}$. The demagnetization, in contrast, is non-linearly related
to both $\Delta T$ and $T_{\text{init}}$ especially in the vicinity of
$T_{\text{C}}$.

Figure 1: Fluence-dependent precession amplitude: (a) Sketch of the sample
structure containing a $20\,\text{nm}$-thick Ni layer and the time-resolved
polar MOKE experiment providing double-pulse excitation with variable pump-
pump delay $\Delta t$ and fluence of the first ($F_{1}$) and the second
($F_{2}$) pump pulse. (b) Fluence-dependent magnetization precession upon a
single excitation for an external field of $350\,\text{mT}$ at $35^{\circ}$
with respect to the sample surface normal.

Figure 2(b) illustrates how the precession amplitude
$A=\Delta\vec{H}_{\text{eff}}\times\vec{M}$ depends on the ”initial”
temperature $T_{\text{init}}$, set by a first pulse[26, 27], and the
temperature change $\Delta T$ set by a second pulse. The blueish color denotes
an expansion-dominated driving of the precession (blue line in Fig. 2(a)) and
reddish color denotes a demagnetization-dominated driving of the precession
starting with an out-of-plane tilting of the magnetization (red line in Fig.
2(a)). The magneto-elastic coupling parameter $b_{1}=-8.24\cdot
10^{6}\,\text{J}\,\text{m}^{-3}$ has been calibrated for a similar Ni film[16]
and the quasi-static expansion $\eta_{\text{qs}}=\alpha\Delta T$ is given by
the ultrafast expansion coefficient $\alpha=2.02\cdot 10^{-5}\,\text{K}^{-1}$
of Ni[28, 29]. The demagnetization $\Delta M(t)$ is approximated by the
temperature-dependent magnetization in thermal equilibrium $M(T)$ given by the
steady state of a microscopic 3-Temperature model (m3TM) [30] with
$T_{\text{C}}=630\,\text{K}$: $\Delta M(t)=M(T_{\text{init}}+\Delta
T)-M(T_{\text{init}})$. Additionally, we use the saturation magnetization
$M_{\text{sat}}=4.0\cdot 10^{5}\,\text{A}\,\text{m}^{-1}$ determined by
vibrating sample magnetometry.

The grey solid line in Fig. 2(c) represents the modelled precession amplitude
$A$ for increasing $\Delta T$ at $T_{\text{init}}=300\,\text{K}$ that
describes the fluence series in Fig. 1(b). Assuming
$0.7\,\text{mJ}\,\text{cm}^{-2}$ to increase the phonon temperature by $\Delta
T_{\text{Ni}}=34.5\,\text{K}$ yields excellent agreement of our model with the
fluence-dependent amplitude (colored symbols) extracted from a fit of the
transient magnetization by an exponential decaying background and a damped
oscillation. With increasing fluence the non-linear increase of the
demagnetization reduces the increase of the precession amplitude up to $\Delta
T_{\text{Ni}}=140\,\text{K}$ where the additional demagnetization overcomes
the additional quasi-static expansion and the precession amplitude starts to
decrease again. Beyond $\Delta T_{\text{Ni}}=270\,\text{K}$ the
demagnetization even dominates over the quasi-static expansion. This path
along increasing $\Delta T$ is indicated by the arrow in Fig. 2(b). The
transition through the white area indicates the new dominant mechanism of
demagnetization which can be seen directly in Fig. 1(b) as the $\pi$ phase
shift of the transients with fluence $F=0.7$ and
$5.8\,\text{mJ}\,\text{cm}^{-2}$, although the remagnetization dominating the
dynamics for high fluences superimposes the oscillations.

Figure 2(b) shows that this fluence dependence of the precession amplitude
strongly depends on $T_{\text{init}}$. With increasing $T_{\text{init}}$,
quasi-static expansion and demagnetization compensate each other ($A=0$) at
smaller $\Delta T$ and finally demagnetization always dominates for
$T_{\text{init}}>450\,\text{K}$. We verify our model by double-pulse
experiments. The temperature rise of the first pulse $\Delta T_{1}\propto
F_{1}$ raises the temperature and thus sets the initial temperature
$T_{\text{init},2}=300\,\text{K}+\Delta T_{1}$ for the second pulse. Figure 3
presents three different double-pulse control scenarios: The control
parameters for the precession amplitude are i) the fluence of the first pulse
$F_{1}$ (Fig. 3(a,b)), ii) the time delay $\Delta t$ between the pulses (Fig.
3(c,d)) and iii) the fluence $F_{2}$ of the second pulse. For $F_{2}$ we show
two alternatives according to the fixed pump fluence $F_{1}$ which either sets
the initial temperature $T_{\text{init,2}}$ above or below the temperature
$T_{\text{comp}}$, where the magneto-acoustic and demagnetization mechanisms
compensate each other as drivers for the precession.

Figure 2: Modelled precession amplitude: (a) Sketch of the counteracting
effect of laser-induced demagnetization (left) and quasi-static expansion
(right) on the effective field driving precession with a $\pi$ phase shift
displayed by the red and blue solid line in the middle, respectively. (b)
Amplitude of the precession $A$ as function of the laser-induced temperature
rise in the phonons $\Delta T$ and the sample temperature before excitation
$T_{\text{init}}$ modelled according to Eq. 1. The arrows visualize the
measurement series in Figs. 1 and 3. (c) The modelled precession amplitude as
function of $\Delta T$ at $T_{\text{init}}=300\,\text{K}$ (solid line)
corresponds to the left arrow in (b) and captures the measured amplitude of
the fluence series in Fig. 1(b) (coloured symbols).

The left column of Fig. 3 compares the MOKE response of the sample to the both
pulses (solid lines) to the response when only the first pulse is active
(dashed lines). The right column displays the difference of these two signals.
The oscillation amplitude $A$ of this difference signal represents the change
that is induced by the second pulse, i.e. the value encoded in Fig. 2(b). The
variation of $F_{1}$ for constant $F_{2}=1.8\,\text{mJ}\,\text{cm}^{-2}$ in
panels (a) and (b) corresponds to a variation of $T_{\text{init},2}$ at
constant time delay and laser-induced temperature rise $\Delta
T_{2}=85\,\text{K}$. In agreement with our model, we observe a sign change of
the oscillation amplitude $A$ in panel (b) when
$T_{\text{init,2}}=T_{\text{comp}}$ for $F_{1}\approx
2.4\,\text{mJ}\,\text{cm}^{-2}$. At this fluence $F_{1}$ the dominant driving
mechanism for the second excitation switches to demagnetization. Therefore,
the effect of the second pulse switches from amplifying to attenuating the
precession, i.e. the amplitudes of the solid and dashed lines in Fig. 3(a) are
nearly equal. Interestingly, the rather weak second pulse
($F_{2}=1.8\,\text{mJ}\,\text{cm}^{-2}$) is able to switch off the precession
induced by a strong first pulse with $F_{1}=4.3\,\text{mJ}\,\text{cm}^{-2}$
(see Fig. 3(a)).

Figure 3: Control of the magnetization precession within double-pulse
excitation by tuning the effect of the second pulse: (b,d,f,h) Difference in
the transient magnetization upon both excitations (solid lines) and only the
first excitation (dashed lines) in panels (a,c,e,f), respectively. (a,b)
Variation of fluence of the first pulse for constant
$F_{2}=1.8\,\text{mJ}\,\text{cm}^{-2}$. (c,d) Variation of the pump-pump delay
for constant $F_{1}=2.8\,\text{mJ}\,\text{cm}^{-2}$ and
$F_{2}=1.8\,\text{mJ}\,\text{cm}^{-2}$. Note that the second pulse is chosen
to pump at the same precession phase at all chosen $\Delta t$ and nonetheless
has an effect on the phase for varying $\Delta t$. (e,f) Variation of the
fluence $F_{2}$ for constant $F_{1}=1.1\,\text{mJ}\,\text{cm}^{-2}$. (g,h)
Variation of the fluence $F_{2}$ for constant
$F_{1}=4.3\,\text{mJ}\,\text{cm}^{-2}$. The difference in panels (b,d,f,h)
demonstrate a tuning of the dominant driving mechanism for the second
excitation, which we utilize to selective control the magnetization precession
upon double-pulse excitation in panels (a,c,e,f).

This selection of the dominant driving mechanism via the temperature before
the second excitation $T_{\text{init},2}$ can also be realized by the pump-
pump delay $\Delta t$ (see Fig. 3(c,d)) utilizing the cooling of Ni via heat
transport into the substrate within hundreds of picoseconds. With increasing
$\Delta t$ for fixed $F_{1}=2.8\,\text{mJ}\,\text{cm}^{-2}$ and
$F_{2}=1.8\,\text{mJ}\,\text{cm}^{-2}$, we observe the transition from
demagnetization to magnetoacoustics as the driver of the precession between
$\Delta t=94$ and $188\,\text{ps}$ (Fig. 3(d)). This is in agreement with our
model, where $T_{\text{init},2}$ decreases with $\Delta t$ starting from its
maximum value of $430\,\text{K}$. We chose $\Delta t$ such that the second
pulse always arrives at a maximum of the precession driven by the first pulse.
Thus the time delay does not have the same role as in a conventional coherent
control scenario. Although the second pulse always interacts with the sample
at the same phase value of the precession, the change of the dominant driving
mechanism of the precession switches the effect of the second pulse from
attenuation to amplification. A substrate with a phononic heat conductivity
larger than glass ($\kappa_{\text{glass}}\approx
1\,\text{W}\,\text{m}^{-1}\,\text{K}^{-1}$), e.g. MgO, would enhance the range
of attenuation and amplification that can be selected by the pump-pump delay.

Figures. 3(e,f) demonstrate a similar tuning of the dominant driving mechanism
by the fluence of the second pulse for a fixed
$F_{1}=1.1\,\text{mJ}\,\text{cm}^{-2}$. The amplitude $A$ driven by the second
pulse increases with increasing $F_{2}$ up to its maximum value for
$F_{2}=2.2\,\text{mJ}\,\text{cm}^{-2}$. A further increase of $F_{2}$ reduces
the amplitude again and even inverts the sign. For the highest fluence,
$F_{2}=4.7\,\text{mJ}\,\text{cm}^{-2}$, the demagnetization-dominated
mechanism for the second pulse fully switches off the precession induced by
the first pulse, which is best seen in Fig. 3(e).

This dependence of the precession amplitude $A$ on the fluence $F_{2}$
drastically depends on the fluence of the first pulse $F_{1}$. Figure 3(g,h)
shows that for $F_{1}=4.3\,\text{mJ}\,\text{cm}^{-2}$, any choice of the
fluence $F_{2}$ drives the precession dominantly by demagnetization, and hence
reduces the precession amplitude when the second pulse excites one precession
period after the first pulse that drove the precession via magneto-acoustics.
This is again in full in agreement with the model (see arrow in Fig. 2(b)).

In summary, we demonstrated that a transition of the dominant driving
mechanism of a second excitation can be achieved by the fluences of both
pulses and the pump-pump delay in a double-pulse excitation scheme. Our model
and experiment confirm that the demagnetization and quasi-static expansion
counteract each other as precession drivers if the magneto-elastic coupling is
negative ($b_{1}<0$). The change of the dominant driving mechanism from
magneto-elastic coupling to demagnetization occurs at a fluence-dependent
compensation temperature $T_{\text{comp}}$ below the Curie temperature
$T_{\text{C}}$. At this point the effect of the second pulse on the precession
switches from amplification to attenuation. We demonstrate that this mechanism
can be exploited in various ways thanks to the non-linear fluence-dependent
demagnetization mechanism, enabling coherent control beyond the standard
superposition of waves.

We acknowledge the DFG for financial support via No. BA 2281/11-1 and Project-
No. 328545488 – TRR 227 (projects A2 and A10). We thank Alexander von Reppert
from the University of Potsdam for stimulating discussions and Juan-Carlos
Rojas-Sánchez from Institute Jean Lamour in Nancy, France for determining the
saturation magnetization.

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that the chromaticity introduced by reflections at the edge of the REACH
ground plane, if properly modelled, would not severely affect the ability of
the pipeline to recover the cosmological signal. Bottom-right: Plot showing
the result of running the data pipeline when a sinusoidal systematic arising
from the presence of the 6 m cable connecting the spiral antenna feed point to
the receiver has been introduced in the data. The additive systematic signal
is shown as a black-solid line in this plot. In the simulated analysis we
included a sinusoidal model to fit for this systematic signal simultaneously
with the foregrounds and the cosmological signal. This result shows that a
detection of the true signal could be achieved in this case.