# Accelerating bianchi type dark energy cosmological model with cosmic string
in $f(T)$ Gravity

S. H. Shekh1111Email<EMAIL_ADDRESS>V. R. Chirde 2222Email:
<EMAIL_ADDRESS>1 Department of Mathematics, S.P.M. Science and
Gilani Arts and Commerce College, Ghatanji, Yavatmal, Maharashtra-445301,
India.  2 Department of Mathematics, G.S.G. Mahavidyalaya, Umarkhed-445206,
India.

###### Abstract

Abstract: In this paper, we have investigated some features of anisotropic
accelerating Bianchi type-I cosmological model in the presence of two non-
interacting fluids, i.e. one usual string and other dark energy fluid towards
the gravitational field equations for the linear form of $f(T)$ gravity, where
T be the torsion. To achieve a physically realistic solution of the field
equations, we have considered an exact matter dominated volumetric power law
expansion. The aspects of derived model are discussed with the help of
solution and make the model at late times turn out to be flat Universe. Also,
observed that the model has initially non-singular and stable while for whole
expansion it is unstable. At an initial expansion $0.01\leq t\leq 0.05$ when
the Universe start to expand to infinite expansion $t>1.73$ (break away from
for small interval of cosmic time $0.06\leq t\leq 1.73$), there is a matter
dominated era while for $0.06\leq t\leq 1.73$ it approaches the quintessence
region. Also, the string exists in the early universe which is in agreement
with constraints of CMBR data but with the expansion the string phase of the
Universe disappears, i.e. we have an anisotropic fluid of particles.

###### pacs:

95.36.+x; 98.80.Cq; 04.50.Kd.

Keywords: Bianchi type space-time; dark energy; cosmic string; modified
gravity.

##  I Introduction

The Cosmic Microwave Background radiation as seen in data from satellites such
as the Wilkinson Microwave Anisotropy Probe (Knop et al. 2003) and from
observations of Large Scale Structure (Tegmark et al. 2004a, 2004b) of the
universe confirmed that the problem of cosmic acceleration one. Basically, two
kinds of alternative explanations have been proposed for this unexpected
observational phenomenon. One is, the problem lies in detecting an exotic type
of unknown repulsive force (an exotic energy with large negative pressure),
termed as dark energy which is responsible for the accelerating phase of the
Universe and other, by modifying the Einstein equation called Modified Gravity
which is of great importance because it can successfully explain the rotation
curve of galaxies and the motion of galaxy clusters in the universe.
The detection of dark energy would be a new clue to an old puzzle: the
gravitational effect of the zero-point energies of particles and fields
(Zel’dovich 1968). The total with other energies, that are close to
homogeneous and nearly independent of time, acts as dark energy. The paramount
characteristic of the dark energy is a constant or slightly changing energy
density as the Universe expands, but we do not know the nature of dark energy
very well (Turner & Huterer 2007; Bartelmann 2010). Dark energy has been
conventionally characterized by the Equation of State parameter
$\omega=p/\rho$, where $\omega$ is not necessarily constant. The $\omega$ lies
close to $\mathrm{-}$1: it would be equal to (standard cosmology), a little
bit upper than $\mathrm{-}$1 (the quintessence dark energy) or less than
$\mathrm{-}$1 (phantom dark energy). The possibility $\omega<<-1$ is ruled out
by current cosmological data. Akarsu & Kilinc (2010) have studied locally
rotationally symmetric Bianchi type-I cosmological models with a constant
deceleration parameter in the presence of anisotropic dark energy and a
perfect fluid. They have considered phenomenological parameterization of
minimally interacting dark energy in terms of its equation of state parameter
and time-dependent skewness parameters. Pawar & Solanke (2014a) gives an exact
kantowski-sachs anisotropic dark energy cosmological models in Brans Dicke
theory of gravitation. Pawar et al. (2014b) investigated magnetized dark
energy cosmological models with time dependent cosmological term in Lyra
geometry. Mahanta et al. (2014) constructed dark energy anisotropic Bianchi
type-III cosmological models with a variable equation of state parameter in
Barber’s second self-creation theory of gravitation using the special law of
variation of Hubble’s parameter that yields a constant value of the
deceleration parameter and obtained two different models in which first model
the value of equation of state parameter is in good agreement with the recent
observations of type Ia supernovae data with cosmic microwave background
radiation anisotropy and galaxy clustering statistics. Recently Aditya et al.
(2019) investigated the dark energy phenomenon by studying the Tsallis
holographic dark energy within the framework of Brans Dicke scalar–tensor
theory of gravity where the Brans and Dicke scalar field is a logarithmic
function of the average scale factor and Hubble horizon as the IR cutoff and
very recently Singh et al. (2020) examined dark energy cosmological model in
modified scale covariant theory of gravitation. The dynamical dark energy
models are classified into two different categories: (i) the scalar fields
including Quintessence, Phantom, Quintom, K-essence, Tachyon, Dilaton, and so
forth; (ii) the interacting models of dark energy such as the Chaplygin gas
models (Sarkar 2014), Agegraphic (Wei & Cai 2008) and Holographic (Thomas
2002; Setare 2007) models. In recent years, an interesting observation is made
to determine the nature of dark energy in quantum gravity which is termed as
holographic dark energy using principle of holographic dark energy. This
principle (the degrees of freedom in a bounded system should be finite and
does not scale by it volume but with its boundary era) was first put forwarded
by Hooft (1993) in the context of black hole physics. Latter on Fischler &
Susskind (1998) have proposed a new version of the holographic principle, viz.
at any time during cosmological evolution, the gravitational entropy within a
closed surface should not be always larger than the particle entropy that
passes through the past light-cone of that surface. In the context of the dark
energy problem, though the holographic principle proposes a relation between
the holographic dark energy density $\rho_{\Lambda}$ and the Hubble parameter
$H$as $\rho_{\Lambda}=H^{2}$ , it does not contribute to the present
accelerated expansion of the universe along with Granda & Olivers (2008)
proposed a holographic density of the form $\rho_{\Lambda}\cong\alpha
H^{2}+\beta H$ , where $\alpha$, $\beta$ are constants which must satisfy the
restrictions imposed by the current observational data.
There are various modified gravity, One replaces the Ricci scalar R in the
Einstein-Hilbert action by an arbitrary function of R belongs to the well-
known $f(R)$ modified gravity (Azadi et al. 2008; Sharif & Yousaf 2014; Chirde
& Shekh 2016a; 2017). Another one that has gained much regard in the last few
years is Gauss-Bonnet gravity say $f(G)$ theory of gravity (Nojiri & Odintsov
2005, 2007; Cognola 2007; Fayaz et al. 2015; Abbas et al. 2015; Sharif &
Fatima 2016) where $f(G)$ is a generic function of the Gauss-Bonnet invariant
G. Also the theory which combines Ricci scalar and Gauss-Bonnet scalar called
$f(R,G)$ gravity theory (Alvaro & Diego 2012; Shekh & Chirde 2019; Makarenko
et al. 2013; Atazadeh & Darabi 2014; Laurentis & Revelles 2015; Shekh et al.
2020). One the gravitational action includes an arbitrary function of the
Ricci scalar and trace of the stress energy tensor known as $f(R,T)$ gravity.
Several authors such as Sahoo & Mishra (2014), Chirde & Shekh (2015, 2016b),
Pawar & Agrawal (2017) who have investigated the aspects of cosmological
models in this gravity. Among the various modifications of Einstein’s theory,
another one way to look at the theory beyond the Einstein equation is the
Teleparallel Gravity which uses the Weitzenbock connection in place of the
Levi–Civita connection and so it has no curvature but has torsion which is
responsible for the acceleration of the Universe. Some relevant works in this
gravity are the graphical representation of k-essence with the help of
equation of state parameter describe by Sharif & Rani (2011). In $f(T)$
gravity the existence of relativistic stars investigated by Bohmer et al.
(2011). Chirde & Shekh (2014) have discussed some cosmological model with
different sources. Gamal & Nashed (2015) investigated anisotropic models with
two fluids in linear and quadratic forms of $f(T)$gravitational models.
Recently, Bhoyar et al. (2017) discussed stability of accelerating Universe
with linear equation of state parameter in $f(T)$ gravity using hybrid
expansion law.

Bianchi type cosmological models are significant in the sense that these are
homogeneous and anisotropic. Bianchi type space times provide spatially
homogeneous and anisotropic models of the universe as compared to the
homogeneous and isotropic FRW models, from which the course of action of
isotropization of the universe is well thought-out the passage of time. The
simplicity of the field equations and relative ease of the solutions of
Bianchi space times are useful in constructing models of spatially homogeneous
and anisotropic cosmologies. Spatially homogeneous and anisotropic
cosmological models play a significant role in the description of large scale
behavior of the universe and such models have been widely studied many authors
in search of relativistic picture of the early universe. Mohanty et al.
(2003), Pradhan et al. (2011), Kumar & Yadav (2011), Yadav & Saha (2012) are
the some authors who have been extensively investigated an anisotropic Bianchi
type-I, Bianchi type-III, Bianchi type-V dark energy models with the usual
perfect fluid. Recently, Sarkar & Mahanta (2013) have studied a homogeneous
and anisotropic axially symmetric Bianchi type-I universe filled with matter
and holographic dark energy energy components by assuming deceleration
parameter to be a constant, along with the correspondence between the
holographic dark energy models with the quintessence dark energy and
reconstruction of the Quintessence potential and the dynamics of the
quintessence scalar field in an accelerated expansion of the universe. Mishra
et al. (2017) who have investigated the anisotropic behavior of the
accelerating universe with the matter field of two non-interacting fluids
usual string fluid and dark energy fluid and achieve a physically reasonable
clarification using hybrid scale factor, which is generated by a time-varying
deceleration parameter and observed that the string fluid dominates the
universe at early deceleration phase but does not influence nature of cosmic
dynamics significantly at late phase, whereas the dark energy fluid dominates
the universe in present time, which is in accordance with the observations
results. Chirde & Shekh (2018a; 2018b) have investigated the dynamics of
Bianchi type-I space–time filled with two minimally interacting fields, matter
and Holographic dark energy components with volumetric power and exponential
expansion laws and the transition between general relativity and quantum
gravity using quark and strange quark matter respectively towards the
gravitational field equations for the linear form of $f(T)$ gravity and
observed that both models are at late times turn out to be flat Universe, the
power law model has to begin with singular and stable but at spreading out it
is unstable while exponential model is free from any types of singularities
and stable throughout the expansion. In the same way the models represents its
spreading as accelerated with inflationary era in the early and the very late
time of the Universe. For suitably large time resulting model expects that the
anisotropy of the Universe will damp out and the Universe will become
isotropic. Very recently, Pawar et al. (2019) the author who have studied the
modified holographic Ricci dark energy model in $f(R,T)$ theory of gravity by
using anisotropy Bianchi type Universe and observed that the model has no
physical singularity, the Universe is expanding and accelerating
exponentially.

##  II Preliminary definitions and equations of motion of
$\boldsymbol{f}\boldsymbol{(}\boldsymbol{T}\boldsymbol{)}$ gravity:

In this section we give a brief description of $f(T)$ gravity and a detailed
derivation of its field equations. Let us define the notations of the Latin
subscript as these related to the tetrad field and the greek one related to
the space-time coordinates. For a general space-time metric, we can define the
line element as

$dS^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu},$ (1)

where $g_{\mu\nu}$ are the components of the metric which is symmetric and
possesses ten degrees of freedom. One can describe the theory in the space-
time or in the tangent space, which allows us to rewrite the line element
which can be converted to the Minkowski’s description of the transformation
called tetrad (which represent the dynamic fields of the theory), as follows

$dS^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=\eta_{ij}\theta^{i}\theta^{j},$ (2)
$dx^{\mu}=e_{i}^{\mu}\theta^{i},{\rm\;\;\;}\theta^{i}=e_{\mu}^{i}dx^{\mu},$
(3)

where $\eta_{ij}$ is a metric on Minkowski space-time and
$\eta_{ij}=diag[1,-1,-1,-1]$ and $e_{i}^{\mu}e_{\nu}^{i}=\delta_{\nu}^{\mu}$
or $e_{i}^{\mu}e_{\mu}^{j}=\delta_{i}^{j}$. The root of metric determinant is
given by $\sqrt{-g}=\det[e_{\mu}^{i}]=e$. For a manifold in which the Riemann
tensor part without the torsion terms is null (contribution of the Levi-Civita
connection) and only the non-zero torsion terms exist, the Weitzenbocks
connection components are defined as

$\Gamma^{\alpha}{}_{\mu\nu}=e_{i}^{\alpha}\partial_{\nu}e_{\mu}^{i}=-e_{\mu}^{i}\partial_{\nu}e_{i}^{\alpha}.$
(4)

which has a zero curvature but nonzero torsion. The main geometrical objects
of the space-time are constructed from this connection. Through the
connection, the components of the tensor torsion are defined by the
antisymmetric part of this connection as

$T^{\alpha}{}_{\mu\nu}=\Gamma^{\alpha}{}_{\mu\nu}-\Gamma^{\alpha}{}_{\nu\mu}=e_{i}^{\alpha}\left(\partial_{\mu}e_{\nu}^{i}-\partial_{\mu}e_{\mu}^{i}\right),$
(5)

The difference between the Levi-Civita and Weitzenbock connections is a space-
time tensor, and is known as the contorsion tensor:

$K^{\mu\nu}{}_{\alpha}=\left(-\frac{1}{2}\right)\left(T^{\mu\nu}{}_{\alpha}+T^{\nu\mu}{}_{\alpha}-T^{\mu\nu}{}_{\alpha}\right).$
(6)

In order to make more clear the definition of the scalar equivalent to the
curvature scalar of RG, we first define a new tensor $S^{\mu\nu}{}_{\alpha}$,
constructed from the components of the tensors torsion and contorsion tensor
as

$S^{\mu\nu}{}_{\alpha}=\left(\frac{1}{2}\right)\left(K^{\mu\nu}{}_{\alpha}+\delta_{\alpha}^{\mu}T^{\beta\nu}{}_{\beta}-\delta_{\alpha}^{\nu}T^{\beta\mu}{}_{\beta}\right).$
(7)

The torsion scalar $T$ is

$T=T^{\alpha}{}_{\mu\nu}{\rm\;}S^{\mu\nu}{}_{\alpha}.$ (8)

Now, we define the action by generalizing the TG i.e. $f(T)$ theory as

$S=\int\left[T+f(T)+L_{matter}\right]{\rm\;}e{\rm\;}d^{4}x.$ (9)

Here, $f(T)$ denotes an algebraic function of the torsion scalar T. Making the
functional variation of the action (9) with respect to the tetrads, we get the
following equations of motion

$S_{\mu}^{\nu\rho}\partial_{\rho}Tf_{TT}+\left[e^{-1}e_{\mu}^{i}\partial_{\rho}\left(ee_{i}^{\alpha}S_{\alpha}^{\nu\rho}\right)+T^{\alpha}{}_{\lambda\mu}S_{\alpha}^{\nu\lambda}\right]f_{T}+\frac{1}{4}\delta_{\mu}^{\nu}\left(f\right)=4\pi(T_{\mu}^{\nu}{}^{(CS)}+\bar{T}_{\mu}^{\nu}{}^{(DE)}).$
(10)

The field equation (10) is written in terms of the tetrad and partial
derivatives and appears very different from Einstein’s equation.
where $T_{\mu}^{\nu}$ is the energy momentum tensor, $f_{T}=df(T)/dT$,
$f_{TT}=d^{2}f(T)/dT^{2}$ and by setting $f(T)=a_{0}=$ constant the equations
of motion (10) are the same as that of the Teleparallel Gravity with a
cosmological constant, and this is dynamically equivalent to the GR. These
equations clearly depend on the choice made for the set of tetrads.
$T_{\mu}^{\nu}{}^{(CS)}$ and $\bar{T}_{\mu}^{\nu}{}^{(DE)}$ respectively,
denote the contribution to energy–momentum tensor from one-dimensional cosmic
string and dark energy.
In recent years, Cosmologists have taken considerable interest in the study of
cosmic strings due to the challenging problem to determine the exact physical
scenario at very early stages of the formation of the Universe. Since, they
believed that string plays an important role in the description of the
Universe in the early stages of evolution (these arise during the phase
transition after the big-bang explosion as the temperature decreased below
some critical temperature as predicted by grand unified theories) (Kibble
1976). Cosmic string is topologically stable objects that might be found
during a phase transition in the early universe and give rise to density
perturbations leading to the formation of galaxies (Zeldovich 1980). The
innovative work in the formation of the energy-momentum tensor for classical
massive strings was done by Letelier (1983), who used this idea to obtain a
cosmological solution for Bianchi-I and Kantowski-Sachs space-times by
considering the massive strings to be formed by geometric strings with
particle attached along its extension. As in our investigations, one matter
content is string cloud fluid. So that the energy momentum tensor for cloud
string is defined as

$T_{\mu}^{\nu}{}^{(CS)}=(\rho+p){\rm\;}u^{\nu}u_{\mu}-p{\rm\;}g_{\mu}^{\nu}+\lambda
x^{\nu}x_{\mu},$ (11)

where $u^{\nu}$ is the four velocity of the string cloud, $x^{\nu}$ is the
normal space-like four-vector, the pressure p for this fluid is taken to be
isotropic, $\rho$ and $\lambda$ are the proper energy density for a cloud of
strings and the tension density of the string cloud respectively. Here we
consider the string source is along z-axis, which is the axis of symmetry, an
orthonormalizsation of $u^{\nu}$ and $x^{\nu}$ is given by

$u^{\nu}u_{\nu}=1,u^{\nu}x_{\nu}=0,x^{\nu}x_{\nu}=-1,$ (12)

where $u^{\nu}=\left(0,0,0,1\right)$ and $x^{\nu}=\left(A^{-1},0,0,0\right)$.
Saha et al. (2010) investigated Bianchi type-I cosmological model in the
presence of a magnetic flux along with a cosmic string using some tractable
assumptions regarding the parameters entering the model and obtained the
analytical results which are supplemented with numerical and qualitative
analysis describing the evolution of the Universe for different values of the
parameters. Sahoo (2015) have studied the non-singular, non-rotating and
expanding LRS Bianchi type-I cosmological model in modified gravity with
cosmic string and observed that the transition of the universe from
decelerated phase to the accelerated phase at late times in accordance with
the observations of modern cosmology. The existence of the late time
acceleration of the universe with string fluid as source of matter in
anisotropic Heckmann-Suchking space-time by using high red shift $(0.3\leq
z\leq 1.4)$ SNe-Ia data of observed complete magnitude along with their
possible error from Union 2.1 compilation. It is found that the best fit
values for
$\left(\Omega_{m}\right)_{0},{\rm\;\;}\left(\Omega_{\Lambda}\right)_{0},{\rm\;\;}\left(\Omega_{\sigma}\right)_{0}$
and $(q)_{0}$ are 0.2820, 0.7177, 0.0002 & -0.5793 respectively offered
Goswami et al. (2016).

The energy–momentum tensor for isotropic dark energy can be described as

$\bar{T}_{\mu}^{\nu}{}^{(DE)}=diag[-\rho_{\Lambda},{\rm\;\;}p_{\Lambda},{\rm\;\;}p_{\Lambda},{\rm\;\;}p_{\Lambda}{\rm\;}]$
$=diag[-1,{\rm\;}\omega_{\Lambda},{\rm\;}\omega_{\Lambda},{\rm\;}\omega_{\Lambda}{\rm\;}]{\rm\;}\rho_{\Lambda},$
(13)

where $\omega_{\Lambda}$ is the equation of state parameter for dark energy
and $\rho_{\Lambda}$ is the dark energy density.

##  III  Field equations for Bianchi type-I model:

Let us first establish the equations of motion of a set of diagonal tetrads
using the Cartesian coordinate metric, for describing models of Bianchi
type-I, as

$ds^{2}=dt^{2}-A^{2}dx^{2}-B^{2}dy^{2}-C^{2}dz^{2},$ (14)

Let us choose the following set of diagonal tetrads related to the metric (14)

$[e^{\nu}{}_{\mu}]=diag[1,{\rm\;}A,{\rm\;}B,{\rm\;}C].$ (15)

The determinant of the matrix (14) is

$e={\rm\;}ABC.$ (16)

The components of the tensor torsion (5) for the tetrads (15) are given by

$T^{1}{}_{01}=\frac{\dot{A}}{A},T^{2}{}_{02}=\frac{\dot{B}}{B},T^{3}{}_{03}=\frac{\dot{C}}{C}.$
(17)

and the components of the corresponding tensor contorsion are

$K^{01}{}_{1}=\frac{\dot{A}}{A},K^{02}{}_{2}=\frac{\dot{B}}{B},K^{03}{}_{3}=\frac{\dot{C}}{C},S_{3}{}^{30}=\frac{1}{2}\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}\right).$
(18)

The components of the tensor $S_{\mu}{}^{\mu\nu}$, in (7), are given by

$S_{1}{}^{10}=\frac{1}{2}\left(\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right),S_{2}{}^{20}=\frac{1}{2}\left(\frac{\dot{A}}{A}+\frac{\dot{C}}{C}\right),S_{3}{}^{30}=\frac{1}{2}\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}\right).$
(19)

The corresponding torsion scalar (8) is given by

$T=-2{\rm\;}\left(\frac{\dot{A}}{A}\frac{\dot{B}}{B}+\frac{\dot{A}}{A}\frac{\dot{C}}{\dot{C}}+\frac{\dot{B}}{B}\frac{\dot{C}}{C}\right).$
(20)

Now, the field equations in the framework of $f(T)$gravity (10) for a two-
fluid cosmic string (11) and dark energy (13) Bianchi type-I space-time (14)
are obtained as

$f+4f_{T}\left\\{\frac{\dot{A}}{A}\frac{\dot{B}}{B}+\frac{\dot{A}}{A}\frac{\dot{C}}{C}+\frac{\dot{B}}{B}\frac{\dot{C}}{C}-\frac{\alpha^{2}}{2A^{2}}\right\\}=\left(\rho+\rho_{\Lambda}\right),$
(21)
$f+2f_{T}\left\\{\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}+\frac{\dot{A}}{A}\frac{\dot{B}}{B}+\frac{\dot{A}}{A}\frac{\dot{C}}{C}+2\frac{\dot{B}}{B}\frac{\dot{C}}{C}\right\\}+2\left(\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right)\dot{T}f_{TT}=-\left(p+\omega_{\Lambda}\rho_{\Lambda}\right),$
(22)
$f+2f_{T}\left\\{\frac{\ddot{A}}{A}+\frac{\ddot{C}}{C}+\frac{\dot{A}}{A}\frac{\dot{B}}{B}+2\frac{\dot{A}}{A}\frac{\dot{C}}{C}+\frac{\dot{B}}{B}\frac{\dot{C}}{C}\right\\}+2\left(\frac{\dot{A}}{A}+\frac{\dot{C}}{C}\right)\dot{T}f_{TT}=-\left(p+\omega_{\Lambda}\rho_{\Lambda}\right),$
(23)
$f+2f_{T}\left\\{\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+2\frac{\dot{A}}{A}\frac{\dot{B}}{B}+\frac{\dot{A}}{A}\frac{\dot{C}}{C}+\frac{\dot{B}}{B}\frac{\dot{C}}{C}-\frac{\alpha^{2}}{A^{2}}\right\\}+2\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}\right)\dot{T}f_{TT}=-\left(p+\lambda+\omega_{\Lambda}\rho_{\Lambda}\right),$
(24)

where the dot ($\cdot$) denotes the derivative with respect to cosmic time t.
In the particular case where $f(T)=T-2\Lambda$, the equations (21) - (24) are
identical to that of the general theory of relativity. Finally, here we have
four differential equations with eight unknowns namely
$A,{\rm\;}B,{\rm\;}f,{\rm\;}p,{\rm\;}\rho,\lambda,\rho_{\Lambda},\omega_{\Lambda}$.
The solution of these equations is discussed in next section. In the following
we define some kinematical quantities of the space-time. We define average
scale factor and volume respectively as

$R^{3}=V=ABC.$ (25)

Another important dimensionless kinematical quantity is the mean deceleration
parameter which tells whether the Universe exhibits accelerating volumetric
expansion or not is

$q=-1+\frac{d}{dt}\left(\frac{1}{H}\right),$ (26)

for $-1\leq q<0$, $q>0$ and $q=0$ the Universe exhibit accelerating volumetric
expansion, decelerating volumetric expansion and expansion with constant-rate
respectively. The mean Hubble parameter, which expresses the volumetric
expansion rate of the Universe, given as

$H=\frac{1}{3}\left(H_{1}+H_{2}+H_{3}\right),$ (27)

where $H_{1}$, $H_{2}$ and ${\rm\;}H_{3}$ are the directional Hubble parameter
in the direction of x, y and z-axis respectively. Using equations (25) and
(27), we obtain

$H=\frac{1}{3}\frac{\dot{V}}{V}=\frac{1}{3}\left(H_{1}+H_{2}+H_{3}\right)=\frac{\dot{R}}{R}.$
(28)

To discuss whether the Universe either approach isotropy or not, we define an
anisotropy parameter as

$A_{m}=\frac{1}{3}\sum_{i=1}^{3}\left(\frac{H_{i}-H}{H}\right)^{2}.$ (29)

The expansion scalar and shear scalar are defined as follows

$\theta=u_{;\mu}^{\mu}=\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C},$
(30) $\sigma^{2}=\frac{3}{2}H^{2}A_{m}.$ (31)

##  IV Solution of the field equations:

In order to solve the field equations completely, we first assume that the
matter and holographic dark energy components i.e. the energy momentum tensors
of the two sources interact minimally and conserved separately. The energy
conservation equation of the matter leads to

$\left(\dot{\rho}\right)+\frac{\dot{V}}{V}\left(\rho\right)=0.$ (32)
$\rho=\frac{1}{V}.$ (33)

In this section we find exact solutions of field equations using some physical
quantities for linear $f(T)$ gravity i.e.

$f(T)=T.$ (34)

Now subtracting (22) from (23), we get

$\frac{d}{dt}\left(\frac{\dot{A}}{A}-\frac{\dot{B}}{B}\right)+\left(\frac{\dot{A}}{A}-\frac{\dot{B}}{B}\right)\frac{\dot{V}}{V}=0,$
(35)

which on integration gives

$\frac{A}{B}=k_{2}\exp\left[k_{1}\int\frac{dt}{V}\right],$ (36)

where $k_{1}$and$k_{2}$are constants of integration. In view of equation (25),
we write A and B in the explicit form

$A=D_{1}V^{\frac{1}{3}}\exp\left(\chi_{1}\int\frac{1}{V}dt\right),$ (37)
$B=D_{2}V^{\frac{1}{3}}\exp\left(\chi_{2}\int\frac{1}{V}dt\right),$ (38)

where $D_{i}(i=1,\;2)$ and $\chi_{i}(i=1,\;2)$ satisfy the relation
$D_{1}D_{2}^{2}=1$ and $\chi_{1}+2\chi_{2}=0$.

Exact matter dominated power law solution of the field equation

Since field equations are highly nonlinear, an extra condition is needed to
solve the system completely. So, we have chosen the scale factor of the form

$V=R^{3}=t^{b}$ (39)

where b be the any positive constant. The motivation to choose such scale
factor is that the universe has accelerated expansion at present and
decelerated expansion in the past. Here we are considering the minimally
interacting but conserve separately matter and holographic dark energy
components. Using equations (37), (38) and (39), we obtain the metric
potentials as follows:

$A=D_{1}t^{\frac{b}{3}}\exp\left\\{\frac{\chi_{1}}{(1-b)}t^{1-b}\right\\},$
(40)
$B=D_{2}t^{\frac{b}{3}}\exp\left\\{\frac{\chi_{2}}{(1-b)}t^{1-b}\right\\},$
(41)

and

$C=\frac{1}{D_{1}D_{2}}t^{\frac{b}{3}}\exp\left\\{\frac{\chi_{2}}{(1-b)}t^{1-b}\right\\}.$
(42)

From the equations (40) to (42), it is observed that the metric potentials A,
B and C are the product of power and exponential form and increase
indefinitely with the passage of time. With the help of equations (40) to
(42), our spatially homogeneous and anisotropic Bianchi type-I space-time
filled with cosmic string and holographic dark energy fluid within the
framework of $f(T)$ gravity becomes

$ds^{2}=dt^{2}-D_{1}^{2}t^{\frac{2b}{3}}\exp\left\\{\frac{2\chi_{1}}{(1-b)}t^{1-b}\right\\}dx^{2}-D_{2}^{2}t^{\frac{2b}{3}}\exp\left\\{\frac{2\chi_{2}}{(1-b)}t^{1-b}\right\\}dy^{2}-\frac{1}{D_{1}^{2}D_{2}^{2}}t^{\frac{2b}{3}}\exp\left\\{\frac{2\chi_{2}}{(1-b)}t^{1-b}\right\\}dz^{2}.$
(43)

All the metric potentials in the derived model are vanishes at $t=0$. Hence,
the model has no initial singularity. Afterwards, increase indefinitely with
the passage of time, which is in complete agreement with the Big-Bang model of
the Universe and the model is similar to the investigation those of Chirde &
Shekh (2018).

##  V Kinematical parameters:

Spatial volume and Average scale factor,

$V=R^{3}=t^{b}$ (44)

The expansion scalar,

$\theta=\frac{b}{t},$ (45)

The mean Hubble parameter,

$H=\frac{b}{3t}.$ (46)

We observe that the spatial volume and average scale factor both are die out
at $t\to 0$. Thus, no singularity exists at $t\to 0$ in the resultant model.
The model starts evolving with a zero volume at $t\to 0$. The behavior is
shown in FIG-1. The expansion scalar decreases as time increases and the mean
Hubble parameter is initially large at $t\to 0$, and null at $t\to\infty$. The
expansion scalar $\theta\to 0$ as $t\to\infty$ which indicates that the
universe is expanding with increase of time and the rate of expansion
decreases with the increase of time. This suggests that at initial stage of
the universe, the expansion of the model is much more faster and then slows
down for later time i.e. the evolution of the universe starts with infinite
rate, and with expansion, it declines (see FIG-2).

Figure 1: Graphical representation of average scale factor and spatial volume
versus cosmic time with appropriate choice of constant $b=2$.

Figure 2: Graphical representation of Hubble’s parameter and expansion scalar
versus cosmic time with appropriate choice of constant $b=2$.

An anisotropy parameter,

$A_{m}=\frac{18\chi_{2}{}^{2}}{b^{2}t^{2(b-1)}}.$ (47)

The shear scalar,

$\sigma^{2}=\frac{9\chi_{2}{}^{2}}{3t^{2b}}.$ (48)

The anisotropic parameter and the shear scalar both are the inverse function
of time. Thus the nature of the anisotropic parameter is varying with the
evaluation of the universe. Initially both are in high elevation and at an
infinite expansion it is seen that the model has shear free and that of
anisotropy parameter disappears in the model. The behavior of the anisotropy
parameter and shear scalar versus cosmic time t is shown in FIG-3.

Figure 3: Graphical representation of anisotropy parameter and shear scalar
versus cosmic time with appropriate choice of constants $b=2$ and
$\chi_{2}=1$.

##  VI Physical parameters:

The Torsion scalar for the model becomes

$T=\frac{-6\chi_{1}\chi_{2}}{t^{2b}}-\frac{16b\left(\chi_{1}+\chi_{2}\right)}{3t^{b+1}}-\frac{8b^{2}}{9t^{2}}$
(49)

Figure 4: Graphical representation of Torsion scalar versus cosmic time with
appropriate choice of constants $b=2$, $\chi_{1}=-2$ and $\chi_{2}=1$.

The Torsion of the Universe is time dependent and decreases very rapidly for
the interval of cosmic time $1\leq t\leq 15.5$ and for $t>15.5$ it attend a
very small positive value which is nearly equal to zero, the behavior is
clearly shown in FIG-4. The proper energy density for a cloud of string is

$\rho=\frac{1}{t^{b}}.$ (50)

The energy density of Holographic dark energy is,

$\rho_{\Lambda}=\frac{\chi_{2}(\chi_{1}+\chi_{2})}{t^{2b}}+\left(\frac{b\left(\chi_{1}+3\chi_{2}\right)}{t^{b+1}}\right)-\frac{1}{t^{b}}+\frac{2b^{2}}{9t^{2}}.$
(51)

Figure 5: Graphical representation of energy density versus cosmic time with
appropriate choice of constants $b=2$, $\chi_{1}=-2$ and $\chi_{2}=1$.

In power law expansion of the Universe, it is observed that the energy density
is always positive and decreasing function of cosmic time t. At the initial
stage the Universe has infinitely large energy density but with the expansion
of the Universe it declines and at very large expansion, it is null. The
behavior is clearly shown in FIG-5.
The equation of state parameter for Holographic dark energy is,

$\omega_{\Lambda}=\frac{\left(\frac{\chi_{2}\left(2\chi_{1}-6\chi_{2}-\omega\right)}{t^{2b}}+\frac{b\left(2\chi_{1}-6\chi_{2}-\omega\right)}{3t^{b+1}}\right)}{\left(\frac{\chi_{2}(\chi_{1}+\chi_{2})}{t^{2b}}+\left(\frac{b\left(\chi_{1}+3\chi_{2}\right)}{t^{b+1}}\right)-\frac{1}{t^{b}}+\frac{2b^{2}}{9t^{2}}\right)}.$
(52)

Recently a large class of scalar field dark energy models has been given
including Quintessence $(\omega_{\Lambda}>-1)$, Phantom
$(\omega_{\Lambda}<-1)$ and Quinton (which can cross from the Phantom region
to the quintessence region). The Quinton scenario of dark energy is designed
to understand the nature of dark energy with $\omega_{\Lambda}$ cross $-1$.
Setare & Saridakis (2015) have studied the dark energy models with the
equation of state parameter across ($-1$), which gives a concrete
justification for the Quinton paradigm. Some other limits of equation of state
parameter are obtained from observational results that came from SNe-Ia data
and a combination of SNe-Ia data with CMB anisotropy and Galaxy clustering
statistics are $-1.66<\omega_{\Lambda}<-0.62$ and
$-1.33<\omega_{\Lambda}<-0.79$ respectively. The latest result in 2009,
obtained after a combination of cosmological data sets coming from CMB
anisotropy, luminosity distances of high red-shift SNe-Ia, and galaxy
clustering constrain the dark energy equation of state parameter to
$-1.44<\omega_{\Lambda}<-0.92$. In the derived model, the equation of state
parameter is evolving with a positive sign i.e. $\omega_{\Lambda}>0$ where the
model behaves as like matter dominated once at early stages for the interval
$0.01\leq t\leq 0.05$, for the interval $0.06\leq t\leq 1.73$ the model shows
quintessence region while at late times at $t>1.73$ it be present in matter
dominated once, which may be established from the current accelerated
expansion of the universe. From figure-vi, we observed that at the initial
time there is a quintessence region, and at late time it approaches the
cosmological constant scenario. This is a situation in the early universe
where the quintessence-dominated universe (Caldwell 2002) may be playing an
important role for the equation of state parameter. If the present model is
compared with the experimental results mentioned above, one can conclude that
the limit of equation of state parameter provided by Eq. (31) for some
interval of time it may be accumulated with the acceptable range of equation
of state parameter (see Fig-6). This model confirms the high red-shift
supernova experiment.

Figure 6: Graphical representation of Equation of State parameter versus
cosmic time with appropriate choice of constants $b=2$, $\chi_{1}=-2$,
$\chi_{2}=1$ and $\omega=1/3$.

The tension density of the string cloud is

$\lambda=\left[\frac{\chi_{2}}{t^{b}}+\frac{b}{3t}\right]\left\\{\left.\frac{\omega+4\chi_{2}-2\chi_{1}}{t^{b}}\right\\}-\frac{\omega}{t^{b}}-2\frac{\chi_{1}}{t^{b}}\left[\frac{\chi_{1}}{t^{b}}+\frac{b}{3t}\right]\right..$
(53)

As the cosmologists have taken considerable interest in the study of cosmic
strings. Since, they believed that string plays an important role in the
description of the Universe in the early stages of evolution i.e. arise during
the phase transition after the big bang explosion as the temperature decreased
below some critical temperature as predicted by grand unified theories also
which is a topologically stable objects that might be found during a phase
transition in the early universe, Letelier (1983) pointed out that tension
density of the string cloud may be positive or negative. In our investigation
it is initially positive hence the strings exist in the early universe
occupying the small universe, which is in agreement with constraints of CMBR
data but with the expansion of the Universe it gradually decreases and remain
present with negative value i.e. $\lambda<$ 0 hence, with the expansion the
string phase of the Universe disappears, i.e. we have an anisotropic fluid of
particles, this behavior is clearly shown in FIG-7.

Figure 7: Graphical representation of tension density of the string cloud
versus cosmic time with appropriate choice of constants $b=2$, $\chi_{1}=-2$,
$\chi_{2}=1$ and $\omega=1/3$.

Stability factor of the Universe,

For the stability of corresponding solutions of the derived Universe, we
should check that our Universe is physically acceptable. For this, firstly it
is required that the velocity of sound should be less than velocity of light
i.e. within the range $0<\upsilon_{s}=\frac{\partial p}{\partial\rho}$.

In our derived model, the sound speeds is obtained as

$\upsilon_{s}=\frac{\left(\frac{2b\chi_{2}(2\chi_{2}+\omega)}{t^{2b+1}}+\frac{b(b+1)(2\chi_{2}+\omega)}{3t^{b+2}}\right)}{\left(\frac{2b\chi_{2}^{2}}{t^{2b+1}}+\frac{b}{t^{b+1}}-\frac{b\chi_{2}(b+1)}{t^{b}}-\frac{2b^{2}}{9t^{3}}\right)}.$
(54)

Figure 8: Graphical representation of stability factor of the Universe versus
cosmic time with ape choice of constants $b=2$, $\chi_{1}=-2$, $\chi_{2}=1$
and $\omega=1/3$.

From the FIG-8, it is observed that initially for $0<t\leq 0.90$ stability
factor $\upsilon_{s}>0$ and with the expansion $\upsilon_{s}<0$. Hence the
Universe initially stable but with expansion it is unstable. Therefore, on the
basis of above discussions and analysis, our corresponding solutions are
physically acceptable at initially for small interval of cosmic time.

##  VII Conclusion

The Torsion of the Universe is time varying and falls very rapidly for $1\leq
t\leq 15.5$ and for $t>15.5$ it attends a very small positive value which is
nearly equal to zero. The energy density is always positive hence the model is
realistic, at an initial stage the Universe has infinitely large energy
density but with the expansion of the Universe it declines and at very large
expansion, it is null.

The Equation of State parameter is evolving with a positive sign for this
case, the model behaves as like matter dominated once at early stages at an
interval $0.01\leq t\leq 0.05$, for the interval $0.06\leq t\leq 1.73$ the
model shows quintessence region which may be established from the current
accelerated expansion of the universe while at late times at $t>1.73$ it is
present in matter dominated once. At the initial time there is a quintessence
region, and at late time it is baryonic matter. This is a situation in the
early universe where the quintessence-dominated universe may be playing an
important role for the equation of state parameter. If the present model is
compared with the experimental results, it may be accumulated with the
acceptable range of equation of state parameter. This model confirms the high
red-shift supernova experiment.

Initially the cosmic string has positive, hence the strings exist in the early
universe occupying the small universe, which is in agreement with constraints
of CMBR data but with the expansion of the Universe it gradually decreases and
remain present with negative value i.e. $\lambda<$ 0 hence, with the expansion
the string phase of the Universe disappears, i.e. we have an anisotropic fluid
of particles.

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# A nanoparticle stored with an atomic ion in a linear Paul trap

Dmitry S. Bykov,1∗ Lorenzo Dania,1† Florian Goschin,1 Tracy E. Northup1

1Institut für Experimentalphysik, Universität Innsbruck
Technikerstraße 25, 6020 Innsbruck, Austria
∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
†Present address: Photonics Laboratory, ETH Zürich, CH-8093 Zürich,
Switzerland

> Radiofrequency traps are used to confine charged particles but are only
> stable for a narrow range of charge-to-mass ratios. Here, we confine two
> particles—a nanoparticle and an atomic ion—in the same radiofrequency trap
> although their charge-to-mass ratios differ by six orders of magnitude. The
> confinement is enabled by a dual-frequency voltage applied to the trap
> electrodes. We introduce a robust loading procedure under ultra-high vacuum
> and characterize the stability of both particles. It is observed that slow-
> field micromotion, an effect specific to the dual-field setting, plays a
> crucial role for ion localization and will be important to account for when
> engineering controlled interactions between the particles.

## Introduction

The confinement of charged particles in radiofrequency (RF) traps is a
fundamental technique for mass spectrometry [1], cold molecular chemistry [2],
and quantum information processing [3, 4, 5] and has wide-ranging applications
from the study of condensed-matter material properties [6] to the detection of
millicharged dark matter [7]. Typically, a trap is configured to levitate a
specific object with a predefined charge-to-mass ($Q/m$) ratio. However,
certain applications require interactions between different objects in the
same trap. Examples include highly controlled reactions between cold molecular
ions [8], sympathetic cooling of one ion species with another [9, 10], and
quantum logic spectroscopy [11, 12, 13, 14].

While the $Q/m$ selectivity of RF traps is advantageous for certain
applications, such as mass spectrometry, it limits the range of objects that
can be confined simultaneously in the same trap. To overcome this limitation
and trap two species with very different $Q/m$ ratios, a Paul trap driven by
two voltage sources has been proposed [15, 16, 17], targeting antihydrogen
synthesis [15, 17] and sympathetic cooling of megadalton particles, including
nanoparticles or viruses [16]. In this scenario, the frequency and amplitude
of each voltage source are tuned to provide confinement for a particle with a
specific $Q/m$ ratio without compromising the stability of the second species.

Here, we experimentally demonstrate a dual-frequency linear Paul trap driven
by $17.5\text{\,}\mathrm{MHz}$ and $7\text{\,}\mathrm{kHz}$ voltage sources.
This approach allows us to confine a silica nanoparticle and atomic calcium
ions in the same RF trap, thus creating a hybrid system consisting of an
ultra-high-quality-factor mechanical oscillator [18] and an atomic qubit. The
trapped objects differ by six orders of magnitude in their $Q/m$ ratio and by
eight orders of magnitude in mass.

## Theoretical background

Let us examine the conditions for stable trapping of a particle—either a
nanoparticle or an atomic ion—in the quadrupole electric potential $\Phi$ of a
linear Paul trap driven by two voltage sources. In a coordinate system with
the origin at the trap’s geometric center, the potential in the $xy$ plane
orthogonal to the trap axis can be written as

$\Phi(x,y)=\left(V_{\text{slow}}\cos{\Omega_{\text{slow}}t}+V_{\text{fast}}\cos{\Omega_{\text{fast}}t}\right)\frac{x^{2}-y^{2}}{2r_{0}^{2}},$
(1)

where $r_{\mathrm{0}}=$0.9\text{\,}\mathrm{mm}$$ is the characteristic
distance from the electrodes to the trap axis, $\Omega_{\text{slow}}$ and
$\Omega_{\text{fast}}$ are the two frequencies of the drive voltages, and
$V_{\text{slow}}$ and $V_{\text{fast}}$ are their amplitudes. First, we focus
on the motion of a nanoparticle along one axis; our goal is to compare the
influence of the slow and fast components. The equation of motion along $x$ is

$m_{(\mathrm{n})}\ddot{x}=-Q_{(\mathrm{n})}(V_{\mathrm{slow}}\cos{\Omega_{\mathrm{slow}}t}+V_{\mathrm{fast}}\cos{\Omega_{\mathrm{fast}}t})\frac{x}{r_{0}^{2}},$
(2)

where $m_{(\mathrm{n})}$ and $Q_{(\mathrm{n})}$ are the mass and charge of the
nanoparticle. If we neglect the fast field and work in the pseudopotential
approximation, the oscillation frequency in the effective trap potential can
be written as [19]

$\omega_{(\mathrm{n})}|_{\Omega_{\text{slow}}}=\frac{\Omega_{\text{slow}}q_{(\mathrm{n})}}{2\sqrt{2}},$
(3)

where we have introduced the stability parameter

$q_{(\mathrm{n})}=\frac{2Q_{(\mathrm{n})}V_{\mathrm{slow}}}{m_{(\mathrm{n})}r_{0}^{2}\Omega_{\mathrm{slow}}^{2}}.$
(4)

If, on the other hand, we neglect the slow field, the resonance frequency can
be expressed as a fraction of the frequency calculated in Eq. 3:

$\omega_{(\mathrm{n})}|_{\Omega_{\text{fast}}}=\frac{\Omega_{\text{slow}}}{\Omega_{\text{fast}}}\frac{V_{\mathrm{fast}}}{V_{\mathrm{slow}}}\omega_{(\mathrm{n})}|_{\Omega_{\text{slow}}}\approx
10^{-2}\omega_{(\mathrm{n})}|_{\Omega_{\text{slow}}},$ (5)

where the approximation in the final step is based on typical voltages used in
the experiment. From Eq. 5, we conclude that the fast field acts only as a
perturbation to the motion of the nanoparticle, which is confined by the slow
field, and that no special conditions are imposed on the field parameters.

Next, we turn to the equation of motion of an ion in the dual-frequency field:

$m_{(\mathrm{i})}\ddot{x}=-$\mathrm{\text{ $e$
}}$(V_{\text{slow}}\cos{\Omega_{\text{slow}}t}+V_{\text{fast}}\cos{\Omega_{\text{fast}}t})\frac{x}{r_{0}^{2}},$
(6)

where $m_{(\mathrm{i})}$ is the mass and $\mathrm{\text{ $e$ }}$ the
elementary charge. Under the condition
$\Omega_{\text{slow}}\ll\Omega_{\text{fast}}$, the slow field acts as a slowly
varying DC force, and it is possible to map the ion’s equation of motion to
the Mathieu equation

$\ddot{x}+(a_{\mathrm{eff}}+2q_{\mathrm{(i)}}\cos{2t})x=0$ (7)

for the stability parameters

$a_{\mathrm{eff}}=\frac{4$\mathrm{\text{ $e$
}}$V_{\text{slow}}}{m_{\mathrm{(i)}}r_{0}^{2}\Omega^{2}_{\text{fast}}},$ (8)
$q_{\mathrm{(i)}}=\frac{2$\mathrm{\text{ $e$
}}$V_{\text{fast}}}{m_{\mathrm{(i)}}r_{0}^{2}\Omega^{2}_{\text{fast}}}.$ (9)

For $q_{\mathrm{(i)}}\ll 1$, the stability condition for solutions of the
Mathieu equation is approximately [4]

$|a_{\mathrm{eff}}|<q_{\mathrm{(i)}}^{2}/2.$ (10)

Thus, Eq. 10 imposes conditions on the fast and slow fields with which co-
trapping of a nanoparticle and an ion is possible.

## Experimental approach

Figure 1: Experimental setup. (a) Dual-frequency drive of the linear Paul
trap. (b) Schematic of the procedure for trapping a nanoparticle (NP) and an
ion. Along the $z$ axis, DC endcap electrodes are indicated in gray. Not
shown: two pairs of compensation electrodes for displacement in the $xy$
plane. (c) Camera image of a 40Ca+ ion and a nanoparticle confined in the same
Paul trap. (d) A second image in which two ions are confined with a
nanoparticle.

We implement the dual-frequency trap by applying a voltage oscillating at
$\Omega_{\text{fast}}=$17.5\text{\,}\mathrm{MHz}$$ to one pair of RF
electrodes and a second voltage at
$\Omega_{\text{slow}}=$7\text{\,}\mathrm{kHz}$$ to a second pair. (A similar
configuration is described in Ref. [20] for an experiment with 40Ca+ ions.)
The trap schematic is shown in Fig. 1a. The fast voltage signal at
$\Omega_{\text{fast}}$ is provided by a low-noise function generator
(Rohde&Schwarz SMB100B) and preamplified by a high-power amplifier (Mini-
Circuits LZY-22+). We supply the pre-amplified signal to a tap of the
inductive coil of a resonant LC circuit in which the trap electrodes act as a
capacitance. In the case of the slow voltage at $\Omega_{\text{slow}}$, a
different function generator drives a low-noise, high-voltage amplifier (Trek
PZD700A). The internal resistance of the slow voltage source and a
$4.7\text{\,}\mathrm{nF}$ capacitor act as a low-pass filter, effectively
grounding the fast signal. The slow field sees this capacitor as an open
circuit. Similarly, the inductor of the resonant circuit acts as a high-pass
filter, effectively grounding the slow signal.

As a first step, a nanoparticle is loaded into the trap under ultra-high
vacuum via laser-induced acoustic desorption (LIAD) [21, 22] combined with
temporal control of the Paul-trap potential [23]. The nanoparticle source is a
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick aluminum foil with silica
nanospheres deposited on the front side; the spheres have a nominal diameter
of $300\text{\,}\mathrm{nm}$. A pulsed ablation laser (4 mJ, 5 ns) is focused
on the back side of the foil (Fig. 1b). The geometry of the trap and the
source are as described in Ref. [23], with the exception that here, the
distance between DC endcap electrodes is $3.4\text{\,}\mathrm{mm}$. At this
stage, only the slow voltage at $\Omega_{\text{slow}}$ is applied to the trap
electrodes, with amplitude $V_{\text{slow}}$ =
$1.4\text{\,}\mathrm{k\mathrm{V_{pp}}}$. The endcap voltage is
$400\text{\,}\mathrm{V}$.

After loading, we cool the nanoparticle’s motion using electrical feedback
based on optical detection at $780\text{\,}\mathrm{nm}$. There are two
differences with respect to the protocol of Ref. [24]: First, the
nanoparticle’s position along the $x$, $y$, and $z$ axes is detected with a
confocal setup implemented with fiber-coupled avalanche photodiodes (APDs)
[25, 22, 26, 27]. Second, the $x$\- and $y$-axis feedback signals are combined
and sent to a feedback electrode mounted next to the trap, but the
$z$-feedback signal is sent to the lower endcap electrode through a high-pass
filter. Feedback cooling operates continuously during the subsequent steps.

Next, we increase the nanoparticle charge $Q_{(\mathrm{n})}$. Recall from Eq.
10 that the values of $V_{\text{slow}}$ compatible with co-trapping an ion are
bounded from above; a higher value of $Q_{(\mathrm{n})}$ allows the stiffness
of the nanoparticle’s trap to be maintained while $V_{\text{slow}}$ is reduced
(see Eqs. 3 and 4). When the ablation laser is directed with a pulse energy of
$1\text{\,}\mathrm{mJ}$ to a pure calcium target near the trap (Fig. 1b), it
is observed that $Q_{(\mathrm{n})}$ increases and saturates at around
$300\text{\,}\mathrm{\text{ $e$ }}$ in the absence of a fast field. The
presence of a fast field allows even higher values of $Q_{(\mathrm{n})}$ to be
obtained during this process. Thus, at this point we introduce the fast field
with $V_{\text{fast}}\approx$1.5\text{\,}\mathrm{k\mathrm{V_{pp}}}$$. One to
three pulses of the ablation laser corresponds to an increase of a few
elementary charges; we periodically interrupt this stepwise process to reduce
$V_{\text{slow}}$ such that the $q_{\text{(n)}}$ parameter remains below 0.9.
We stop the charging process at an amplitude
$V_{\text{slow}}=$160\text{\,}\mathrm{\mathrm{V_{pp}}}$$, for which
$Q_{(\mathrm{n})}\approx$800\text{\,}\mathrm{\text{ $e$ }}$$ is reached.

Once the nanoparticle has been localized and its charge increased, we load a
40Ca+ ion into the dual-frequency trap. First, all laser beams required for
ion loading, cooling, and detection are aligned to the geometric center of the
trap. Here, the trapped nanoparticle serves as a scattering target onto which
the attenuated beams are focused. Next, the amplitude of the fast voltage is
set to $2.5\text{\,}\mathrm{k\mathrm{V_{pp}}}$. The voltage on one of the
trap’s two pairs of compensation electrodes is then increased such that the
nanoparticle is displaced in the $xy$ plane, making room for the ion at the
trap center. (As the ion’s charge is three orders of magnitude smaller than
the nanoparticles’s charge, its equilibrium position is less affected by the
compensation field.) The nanoparticle-detection optics and the feedback-
cooling parameters are adjusted to account for this new position. Finally, we
operate the ablation laser, still focused on the calcium target, at a pulse
energy of $0.5\text{\,}\mathrm{mJ}$ in conjunction with diode lasers at
$375\text{\,}\mathrm{nm}$ and $423\text{\,}\mathrm{nm}$ for two-step isotope-
selective photoionization [28, 29]. Once an ion is loaded, it is detected and
Doppler-cooled via fluorescence on the $397\text{\,}\mathrm{nm}$
$4^{2}\text{S}_{1/2}\leftrightarrow 4^{2}\text{P}_{1/2}$ transition, with
repumping on the $866\text{\,}\mathrm{nm}$ $4^{2}\text{P}_{1/2}\leftrightarrow
3^{2}\text{D}_{3/2}$ and $854\text{\,}\mathrm{nm}$
$4^{2}\text{P}_{3/2}\leftrightarrow 3^{2}\text{D}_{5/2}$ transitions. All
three diode-laser frequencies are locked to a wavemeter.

To verify that a nanoparticle and an atomic ion are co-trapped, we illuminate
them with the $397\text{\,}\mathrm{nm}$, $866\text{\,}\mathrm{nm}$, and
$854\text{\,}\mathrm{nm}$ laser fields and capture their image using an
electron multiplying charge-coupled device (EM-CCD) camera. An example image
is shown in Fig. 1c: nanoparticle and ion are separated by
$55(10)\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The laser beams are focused on
the ion, and the nanoparticle is in the tail of the Gaussian intensity
profile; the camera detects elastically scattered light from the nanoparticle.
Only $397\text{\,}\mathrm{nm}$ light is used for imaging, and an optical
bandpass filter (Thorlabs FBH400-40) in front of the camera removes other
wavelengths.

## Experimental results

### Co-trapping conditions

Figure 2: Influence of the slow field on the ion. (a) Blue dots: instability
threshold measured with the ion, plotted in terms of the stability parameters
$q_{\text{(i)}}$ and $a_{\text{eff}}$. Green area: Strutt diagram, calculated
from experimental parameters. Insets are camera images of the ion for
different stability regimes; the scale bar is
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (b) Composite image of the ion at
four positions along the $z$ axis for $q_{\text{(i)}}=0.4$,
$a_{\text{eff}}=0.06$.

To determine whether the theoretical description above is consistent with our
demonstration of co-trapping, we characterize the stability of nanoparticle
and ion separately in the dual-frequency Paul trap. First, a nanoparticle is
confined solely with the slow voltage. The fast voltage is then added with
increasing amplitude up to $2.5\text{\,}\mathrm{k\mathrm{V_{pp}}}$ while the
position and oscillation frequency of the nanoparticle are monitored with a
camera and the confocal detection setup, respectively. We find that the
nanoparticle remains trapped for all amplitudes and that the presence of the
fast voltage increases the nanoparticle’s $1\text{\,}\mathrm{kHz}$ frequency
in the $xy$ plane by $10\text{\,}\mathrm{Hz}$ for
$V_{\text{slow}}=$2.5\text{\,}\mathrm{k\mathrm{V_{pp}}}$$. These observations
are in agreement with Eq. 5.

Next, the ion is confined for several values of $V_{\text{fast}}$. At each
setting, $V_{\text{slow}}$ is increased until the ion is expelled from the
trap; this threshold amplitude is identified as
$V_{\text{slow}}^{\text{max}}$. In Fig. 2a, the pair of stability parameters
$(q_{\text{(i)}},a_{\text{eff}})$ is plotted for each pair
$(V_{\text{fast}},V_{\text{slow}}^{\text{max}})$. Superimposed on these data
points is a Strutt diagram [4] calculated from the solutions to the Mathieu
equation, Eq. 7. We see that the experimentally determined thresholds lie at
the edge of the Strutt diagram’s stable region, consistent with our
expectations. The rightmost data point, at $\text{q}_{\text{(i)}}=0.55$,
corresponds to
$(V_{\text{fast}}=$2.5\text{\,}\mathrm{k\mathrm{V_{pp}}}$,V_{\text{slow}}^{\text{max}}=$260\text{\,}\mathrm{\mathrm{V_{pp}}}$)$,
providing an upper bound for the value of $V_{\text{slow}}$ at which ions can
be loaded.

Camera images provide further evidence of the slow field’s influence on ion
confinement: the two insets of Fig. 2a show that an ion is well localized when
the trap is operated deep in the stable region of the Strutt diagram, but that
its position extends over several tens of micrometers at the instability
threshold. This elongation is due to micromotion—not the well-known
micromotion at $\Omega_{\text{fast}}$ in a single-frequency RF trap [19], but
at the second frequency $\Omega_{\text{slow}}$. Figure 2b is a composite of
four images of this slow-frequency micromotion in the stable region, each
obtained for a different position of the ion along the $z$ axis, set by the
endcap voltage. We see that the micromotion amplitude increases as the ion is
displaced further from the origin. This image underscores the importance of
positioning the ion at the slow-micromotion minimum. For future experiments
based on the controlled interaction of two particles in a dual-frequency trap,
it will be crucial to analyze the impact of slow micromotion.

Figure 3: Ion-particle interaction. Circles: fixed nanoparticle positions in
the dual-frequency trap trap. Diamonds: each ion position is calculated from
the corresponding nanoparticle position. Ion-nanoparticle pairs are indicated
with the same color. Calculation parameters: the nanoparticle’s charge is
$800\text{\,}\mathrm{\text{ $e$ }}$, and the ion’s secular frequencies along
the $x$ and $z$ axes are $\omega_{\text{(i)},x}=$4\text{\,}\mathrm{MHz}$$,
$\omega_{\text{(i)},z}=$1\text{\,}\mathrm{MHz}$$.

Having examined the nanoparticle and ion separately, we turn to the question
of where the two particles are located when they are trapped together. Recall,
for example, that in the loading procedure, it was necessary to position the
nanoparticle such that the ion would be loaded at the trap center, to which
the lasers had been aligned. In Fig. 3, we calculate the equilibrium position
of the ion for different positions of the nanoparticle, using typical
experimental parameters. The ion position is determined by the trap potential,
Coulomb repulsion from the charged nanoparticle, and the DC compensation-
electrode voltages $V_{\text{c}1}$ and $V_{\text{c}2}$. Here, fixing the
nanoparticle position as an input parameter in the calculation is equivalent
to fixing $V_{\text{c}1}$ and $V_{\text{c}2}$. The nanoparticle positions are
chosen to lie on the line $($-50\text{\,}\mathrm{\SIUnitSymbolMicro m}$\leq
x\leq$-50\text{\,}\mathrm{\SIUnitSymbolMicro
m}$,y=$0\text{\,}\mathrm{\SIUnitSymbolMicro
m}$,z=$-1\text{\,}\mathrm{\SIUnitSymbolMicro m}$)$, where the small
displacement along $z$ breaks the symmetry of the nanoparticle with respect to
the endcaps, ensuring that the ion will be found in the half-space $z\geq-1$.
In practice, this displacement is at the resolution limit of our imaging
system. Consider the case of a nanoparticle at $x=$\pm
50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$: the ion is found near the origin,
and we infer that the interaction between nanoparticle and ion is negligible
compared to the interaction with the trap. As the nanoparticle is brought
closer to the origin, the ion is displaced along the $z$ axis, indicating that
the particles’ interaction has become significant. Thus, this calculation
provides intuition for how the competing forces are balanced. It also
highlights that there are two ways to trap an ion at the origin, which, as we
have just seen, minimizes its micromotion at $\Omega_{\text{slow}}$: the
nanoparticle must be displaced either in the $xy$ plane or along the $z$ axis.

### Localization and manipulation of ions and nanoparticles

Figure 4: Ion-nanoparticle configurations. The $z$ axis points in the same
direction in all images. (a) EM-CCD images of the $xy$ and (b) $z$
configurations. (c) Starting from an $xy$ pair, we decrease the compensation-
electrode voltages that have been used to displace the nanoparticle from the
trap center. As a result, the nanoparticle shifts towards the center of the
trap, and the ion is repelled along the axis of weakest confinement, which is
the $z$ axis here. (d) The voltage on one endcap is increased to compensate
for this shift; both particles shift away from that endcap. (e) The
compensation-electrode voltages are decreased further until the nanoparticle
lies on the $z$ axis. (f) The endcap voltage is increased further until the
ion is at the origin.

Let us refer to the case of nanoparticle displacement in the $xy$ plane as an
$xy$ pair and to the case of displacement along the $z$ axis as a $z$ pair.
Examples of the two configurations are shown in Figs. 4a and b. The
nanoparticle in an $xy$ pair experiences excess micromotion and is more
sensitive to voltage noise on the trap electrodes than it would be in the $z$
configuration. Thus, it is desirable to rotate an $xy$ pair into a $z$ pair. A
procedure for this transformation is shown in Figs. 4d$-$4f, in which the
voltages on the compensation electrodes and endcap electrodes are adjusted in
four steps. While it is possible to reduce the transformation to two steps,
there is a risk that the ion’s displacement along the $z$ axis exceeds the
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ waist of the Doppler-cooling
beams. Following the rotation to a $z$ pair, we observe that if the ion is
lost from the trap, another ion can be reloaded directly in this
configuration.

We have observed that only a single ion can be loaded in the $z$
configuration, but in the $xy$ configuration, we can co-trap two ions with a
nanoparticle, as shown in Fig. 1d. To achieve this, the energy of the ablation
laser pulse is increased by 10%, to $0.55\text{\,}\mathrm{mJ}$; the other co-
trapping parameters remain unchanged. We have never observed more than two
ions co-trapped with the nanoparticle, but we expect that this should be
possible with improvements to the experimental setup. In particular, when
operated in the co-trapping regime, our trap currently dissipates much more
power than a typical linear Paul trap, which could be improved with a smaller
ion-electrode spacing. Co-trapping multiple ions with a nanoparticle is an
interesting prospect as one can couple the nanoparticle to different
collective motional modes of the trapped ions, which may offer advantages due
to noise suppression [30] or due to the mode frequency.

## Conclusion and outlook

Calcium ions and a charged silica nanoparticle have been simultaneously
confined in a dual-frequency Paul trap. We have studied the ion–nanoparticle
pair stability as function of the two RF amplitudes, developed a reliable
procedure for loading the trap, and demonstrated a method to swap between
different geometric configurations. In-situ charging of the trapped
nanoparticle is a necessary step to reduce the slow-field amplitude to a level
that permits ion trapping. The mismatch of six orders of magnitude between the
particle’s charge-to-mass ratios leads to a differential response to static
electric fields, which we found to be beneficial for tuning the
ion–nanoparticle separation and maximizing ion-loading efficiencies.

Our results lay the groundwork for a hybrid system in which a motional state
of a nanoparticle is coupled to external or internal degrees of freedom of
trapped ions, with prospects for generating quantum-mechanical states of
levitated macroscopic objects [31]. While the center-of-mass motional
frequencies of nanoparticles in ion traps lie in the kilohertz regime, ro-
vibrational modes have megahertz frequencies [32] similar to those of the
ions’ center-of-mass motion, enabling both resonant [33] and dispersive
couplings [34]. Such a hybrid system has been achieved by coupling single
superconducting qubits to a piezoelectric resonator [35], a surface-acoustic-
wave resonator [36], and a bulk-acoustic-wave resonator [37]; the qubit serves
to generate and probe quantum states of motion of the resonator [38]. Building
on these pioneering results, levitated particles offer the novel prospect of
preparing quantum mechanical superpositions with a spatial extent larger than
the mechanical oscillator itself.

## Acknowledgments

We thank Giovanni Cerchiari and Elisa Soave for helpful discussions.

#### Funding:

This research was supported by the Austrian Science Fund (FWF) under grants
I5540, W1259 and Y951.

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