# PDM KG-Coulombic particles in cosmic string rainbow gravity spacetime and a
uniform magnetic field

Omar Mustafa<EMAIL_ADDRESS>Department of Physics, Eastern
Mediterranean University, G. Magusa, north Cyprus, Mersin 10 - Turkey.

###### Abstract

Abstract: Klein-Gordon (KG) particles in cosmic string rainbow gravity
spacetime and a uniform magnetic field are studied in the context of the so
called, metaphorically speaking, position-dependent mass (PDM) settings. We
show that the corresponding KG-equation collapses into a two-dimensional
radial Schrödinger-Coulomb like model. The exact textbook solution of which is
used to find the energies and wave functions of KG-Coulombic particles (both
constant mass and PDM ones). In so doing, we consider, with $y=E/E_{P}$, four
pairs of rainbow functions: (a) $g_{{}_{0}}\left(y\right)=1$,
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{2}}$, (b)
$g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y}$,
(c) $g_{{}_{0}}\left(y\right)=g_{{}_{1}}\left(y\right)=\left(1-\epsilon
y\right)^{-1}$, and (d) $g_{{}_{0}}(y)=(e^{\epsilon y}-1)/\epsilon y$,
$g_{{}_{1}}\left(y\right)=1$. Interestingly, we observe that the first pair in
(a) introduces the Planck energy $E_{p}$ as a maximum possible KG-
particle/antiparticle energy value.

PACS numbers: 05.45.-a, 03.50.Kk, 03.65.-w

Keywords: Klein-Gordon (KG) particles, position-dependent mass, cosmic string
spacetime, rainbow gravity, uniform magnetic field.

##  I Introduction

Rainbow gravity (RG) has attracted research attention over the years R1 ; R2 ;
R3 ; R4 ; R5  as a semi-classical extension of the deformed/doubly special
relativity into general relativity (GR). It suggests that the energy of the
probe particles affects the spacetime background, at the ultra-high energy
regime, and the spacetime metric becomes energy-dependent R5 ; R6 ; R7 ; R8 ;
R9 ; R10 ; R11 ; R12 ; R13 ; R14 ; R15 ; R16 ; R17 . That is, a cosmic string
spacetime metric (in the natural units $c=\hbar=G=1$)

$ds^{2}=-dt^{2}+dr^{2}+\alpha^{2}\,r^{2}d\varphi^{2}+dz^{2},$ (1)

would, under RG, take an energy-dependent form

$ds^{2}=-\frac{1}{g_{{}_{0}}\left(y\right)^{2}}dt^{2}+\frac{1}{g_{{}_{1}}\left(y\right)^{2}}\left(dr^{2}+\alpha^{2}\,r^{2}d\varphi^{2}+dz^{2}\right);\;y=E/E_{p},$
(2)

where $\alpha$ is a constant related to the deficit angle of the conical
spacetime and is defined as $\alpha=1-4G\mu$, $G$ is the Newton’s constant,
$\mu$ is the linear mass density of the cosmic string so that $\alpha<1,E$ is
the energy of the probe particle, and $E_{p}=\sqrt{\hbar c^{5}/G}$ is the
Planck energy. Here, the signature of the line elements (1) and (2) is
$\left(-,+,+,+\right)$. Moreover, the corresponding metric tensor $g_{\mu\nu}$
reads

$g_{\mu\nu}=diag\left(-\frac{1}{g_{{}_{0}}\left(y\right)^{2}},\frac{1}{g_{{}_{1}}\left(y\right)^{2}},\frac{\alpha^{2}\,r^{2}}{g_{{}_{1}}\left(y\right)^{2}},\frac{1}{g_{{}_{1}}\left(y\right)^{2}}\right);\;\mu,\nu=t,r,\varphi,z,$
(3)

and

$\det(g_{\mu\nu})=-\frac{\alpha^{2}\,r^{2}}{g_{{}_{0}}(y)^{2}g_{{}_{1}}(y)^{6}}\Longrightarrow
g^{\mu\nu}=diag(-g_{{}_{0}}(y)^{2},g_{{}_{1}}(y)^{2},\frac{g_{{}_{1}}(y)^{2}}{\alpha^{2}\,r^{2}},g_{{}_{1}}(y)^{2}).$
(4)

where $g_{{}_{0}}\left(y\right)$, $g_{{}_{1}}\left(y\right)$ are the rainbow
functions.

The Planck energy $E_{p}$, in the RG model, is considered to represent a
threshold separating classical from quantum mechanical descriptions to
introduce itself as yet another invariant energy scale alongside the speed of
light. Consequently, rainbow gravity justifies the modification of the
relativistic energy-momentum dispersion relation into

$E^{2}g_{{}_{0}}\left(y\right)^{2}-p^{2}c^{2}g_{{}_{1}}\left(y\right)^{2}=m^{2}c^{4};\;0\leq\left(y=E/E_{p}\right)\leq
1,$ (5)

where $mc^{2}$ is its rest mass energy. Such a modification is significant in
the ultraviolet limit and is constrained to reproduce the standard GR
dispersion relation in the infrared limit so that

$\lim\limits_{y\rightarrow 0}g_{{}_{k}}\left(y\right)=1;\;k=0,1.$ (6)

The effects of such modifications could be observed, for example, in the tests
of thresholds for ultra high-energy cosmic rays R6 ; R13 ; R14 ; R15 , TeV
photons R16 , gamma-ray bursts R6 , nuclear physics experiments R17 . Rainbow
gravity settings have motivated interesting recent studies on the associated
quantum gravity effects. Such studies include, for example, the
thermodynamical properties of black holes R18 ; R19 ; R20 ; R21 ; R211 , the
dynamical stability conditions of neutron stars R22 , thermodynamic stability
of modified black holes R221 , charged black holes in massive RG R222 , on
geometrical thermodynamics and heat engine of black holes in RG R223 , on RG
and f(R) theories R224 , the initial singularity problem for closed rainbow
cosmology R23 , the black hole entropy R24 , the removal of the singularity of
the early universe R25 , the Casimir effect in the rainbow Einstein’s universe
R8 , massive scalar field in RG Schwarzschild metric R26 , five-dimensional
Yang–Mills black holes in massive RG R27 , etc.

On the other hand, the dynamics of Klein-Gordon (KG) particles (i.e., spin-0
mesons), Dirac particles (spin-1/2 fermionic particles), and Duffen-Kemmer-
Peatiau (DKP) particles (spin-1 particles like bosons and photons) in
different spacetime backgrounds in rainbow gravity are studied. For example,
in a cosmic string spacetime background in rainbow gravity, Bezzerra et al. R8
; R81  have studied Landau levels via Schrödinger and KG equations, Bakke and
Mota R28  have studies the Dirac oscillator, they have also studied the
Aharonov-Bohm effect R29 . Hosseinpour et al. R5  have studied the DKP-
particles, Sogut et al. R11  have studied the quantum dynamics of photon,
Kangal et al. R12  have studied KG-particles in a topologically trivial Gödel-
type spacetime in rainbow gravity, and very recently KG-oscillators in cosmic
string rainbow gravity spacetime in a non-uniform magnetic field are studied
by Mustafa R291  (without and with the position-dependent mass (PDM)
settings). In the current proposal, however, we extend such studies and
consider PDM KG-Coulombic particles in cosmic string rainbow gravity spacetime
and a uniform magnetic field.

One should be reminded, nevertheless, that PDM is a metaphoric notion that
emerges as a manifestation of coordinate transformation/deformation that
renders the mass to become effectively position-dependent R30 ; R31 ; R32 ;
R33 ; R34 ; R35 ; R36 ; R37 . PDM concept has been introduced in the study PDM
KG-oscillators in cosmic string spacetime within Kaluza-Klein theory R38 , in
(2+1)-dimensional Gürses spacetime backgrounds R39 , and in Minkowski
spacetime with space-like dislocation R40 . Basically, for the PDM von Roos
Schrödinger Hamiltonian R30 , it has been shown (c.f., e.g., R31 ; R32 ; R33 )
that the PDM momentum operator takes the form
$\hat{p}_{j}(\mathbf{r})=-i[\partial_{j}-\partial_{j}f(\mathbf{r})/4f(\mathbf{r})]\,;\;j=1,2,3,$
where $f\left(\mathbf{r}\right)$ is a positive-valued dimensionless scalar
multiplier. For more details on this issue the reader may refer to R31 ; R33 ;
R38 ; R39 ; R40 ; R401 ; R402 . This assumption would, in turn, allow one to
cast the PDM von Roos kinetic energy operator (using $\hbar=2m=1$ units in the
von Roos Hamiltonian) as
$\hat{T}(\mathbf{r})\psi(\mathbf{r})=-f(\mathbf{r})^{-1/4}(\mathbf{\nabla\,}f(\mathbf{r})^{-1/2})\cdot(\mathbf{\nabla\,}f(\mathbf{r})^{-1/4}\psi(\mathbf{r}))$
(known in the literature as Mustafa-Mazharimousavi’s PDM kinetic energy
operator R32 ). Which suggests that the momentum operator for constant mass
setting, $\hat{p}_{j}-i\,\partial_{j}$, should be replaced by the PDM operator
$\hat{p}_{j}\left(\mathbf{r}\right)$ for PDM settings. We shall use such PDM
recipe in the current study of PDM KG-Coulombic particles in cosmic string
rainbow gravity spacetime and a uniform magnetic field. We shall be interested
in three pairs of rainbow functions: (a) $g_{{}_{0}}\left(y\right)=1$,
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{2}}$, and
$g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y}$,
which belong to the set of rainbow functions $g_{{}_{0}}\left(y\right)=1$,
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{n}}$ (where $\epsilon$ is a
dimensionless constant of order unity) used to describe the geometry of
spacetime in loop quantum gravity R41 ; R42 , (b)
$g_{{}_{0}}\left(y\right)=g_{{}_{1}}\left(y\right)=\left(1-\epsilon
y\right)^{-1}$, a suitable set used to resolve the horizon problem R13 ; R43 ,
and (c) $g_{{}_{0}}(y)=(e^{\epsilon y}-1)/\epsilon y$ and
$g_{{}_{1}}\left(y\right)=1$, which are obtained from the spectra of gamma-ray
bursts at cosmological distances R6 .

The organization of our paper is in order. We discuss, in section 2, PDM KG-
particles in the cosmic string rainbow gravity spacetime (2) and a uniform
magnetic field. We show that the corresponding KG-equation collapses into the
two-dimensional radial Schrödinger-Coulomb equation. In section 3, we discuss
the RG effect (using the above mention rainbow functions sets) on the
spectroscopic structure of KG-Coulombic constant mass particles. We discuss,
in section 4, the effects of rainbow gravity as well as PDM on the energy
levels of a PDM KG-Coulombic particle. Our concluding remarks are given in
section 5.

##  II PDM KG-particles in cosmic string rainbow gravity spacetime and a
uniform magnetic field

In the cosmic string rainbow gravity spacetime background (2), a KG-particle
of charge $e$ in a 4-vector potential $A_{\mu}$ is described (in $c=\hbar=G=1$
units) by the KG-equation

$\frac{1}{\sqrt{-g}}D_{\mu}\left(\sqrt{-g}g^{\mu\nu}D_{\nu}\Psi\right)=m^{2}\Psi,$
(7)

where $D_{\mu}$ is the gauge-covariant derivative given by
$D_{\mu}=\partial_{\mu}-ieA_{\mu}$, and $m$ is the rest mass energy of the KG-
particle. At this point, we may also include position-dependent mass (PDM)
settings (a metaphoric description of deformed coordinates and inherited from
the von Roos Hamiltonian R30  ) using the PDM-momentum operator
$\hat{p}_{j}(\mathbf{r})=-i[\partial_{j}-\partial_{j}\,f(\mathbf{r})/4\,f(\mathbf{r})]$
R31 ; R32 ; R33 ; R38 ; R39 ; R40 ; R401 ; R402 . In this case,
$D_{\mu}\longrightarrow\tilde{D}_{\mu}=D_{\mu}+\mathcal{F}_{\mu}=\partial_{\mu}+\mathcal{F}_{\mu}-ieA_{\mu}$,
where $\mathcal{F}_{\mu}=(0,\mathcal{F}_{r},0,0)$,
$\mathcal{F}_{r}=f^{\prime}\left(r\right)/4\,f\left(r\right)$ and
$f\left(\mathbf{r}\right)=f\left(r\right)$ is only radially dependent. One
should notice that a KG-oscillator is obtained using $f(r)=\exp(2\beta
r^{2})$, where $f\left(r\right)$ is a positive dimensionless scalar
multiplier. Under such new structure our KG-equation (7) now describes PDM KG-
particles in the cosmic string rainbow gravity spacetime and reads

$\frac{1}{\sqrt{-g}}\tilde{D}_{\mu}\left(\sqrt{-g}g^{\mu\nu}\tilde{D}_{\nu}\right)\Psi=m^{2}\Psi\Longrightarrow\frac{1}{\sqrt{-g}}\left(D_{\mu}+\mathcal{F}_{\mu}\right)\sqrt{-g}g^{\mu\nu}\left(D_{\nu}-\mathcal{F}_{\nu}\right)\Psi=m^{2}\Psi.$
(8)

Which, in a straightforward manner, yields

$\left\\{-g_{{}_{0}}\left(y\right)^{2}\partial_{t}^{2}+g_{{}_{1}}\left(y\right)^{2}\left[\partial_{r}^{2}+\frac{1}{r}\partial_{r}-M\left(r\right)+\frac{1}{\alpha^{2}\,r^{2}}\left(\partial_{\varphi}-ieA_{\varphi}\right)^{2}+\partial_{z}^{2}\right]\right\\}\Psi\left(t,r,\varphi,z\right)=m^{2}\Psi\left(t,r,\varphi,z\right),$
(9)

where

$M\left(r\right)=\mathcal{F}_{r}^{\prime}+\frac{\mathcal{F}_{r}}{r}+\mathcal{F}_{r}^{2}=-\frac{3}{16}\left(\frac{f^{\prime}\left(r\right)}{f\left(r\right)}\right)^{2}+\frac{f^{\prime}\left(r\right)}{4rf\left(r\right)}+\frac{f^{\prime\prime}\left(r\right)}{4f\left(r\right)}$
(10)

We now use the substitution

$\Psi\left(t,r,\varphi,z\right)=\exp\left(i\left[\ell\varphi+k_{z}z-Et\right]\right)\psi\left(r\right),$
(11)

in Eq. (9) to obtain

$\left\\{\tilde{E}^{2}+g_{{}_{1}}\left(y\right)^{2}\left[\partial_{r}^{2}+\frac{1}{r}\partial_{r}-M\left(r\right)-\frac{\left(\ell-
eA_{\varphi}\right)^{2}}{\alpha^{2}\,r^{2}}\right]\right\\}\psi\left(r\right)=0,$
(12)

where

$\tilde{E}^{2}=g_{{}_{0}}\left(y\right)^{2}E^{2}-g_{{}_{1}}\left(y\right)^{2}k_{z}^{2}-m^{2}$
(13)

In what follows we shall consider $A_{\varphi}=\frac{1}{2}B_{\circ}r$, which
in turn yields a non-uniform magnetic field
$\mathbf{B}=\mathbf{\nabla}\times\mathbf{A}=B_{\circ}\,\hat{z}$. Consequently,
Eq.(12) becomes

$\left\\{\lambda+\partial_{r}^{2}+\frac{1}{r}\partial_{r}-M\left(r\right)-\frac{\tilde{\ell}^{2}}{r^{2}}+\frac{\tilde{\ell}\,\tilde{B}}{r}\right\\}\psi\left(r\right)=0,$
(14)

where

$\lambda=\frac{g_{{}_{0}}\left(y\right)^{2}E^{2}-g_{{}_{1}}\left(y\right)^{2}\left(k_{z}^{2}+\frac{\tilde{B}^{2}}{4}\right)-m^{2}}{g_{{}_{1}}\left(y\right)^{2}},\;\tilde{\ell}=\frac{\ell}{\alpha},\;\tilde{B}=\frac{eB_{\circ}}{\alpha}.$
(15)

Moreover, with $\psi\left(r\right)=R\left(r\right)/\sqrt{r}$ we obtain

$\left\\{\partial_{r}^{2}-\frac{\left(\tilde{\ell}^{2}-1/4\right)}{r^{2}}-M\left(r\right)+\frac{\tilde{\ell}\,\tilde{B}}{r}+\lambda\right\\}R\left(r\right)=0.$
(16)

Under such spacetime and magnetic field structures, we shall consider two
types of KG-particles: constant mass and PDM ones.

##  III Constant mass KG-particles in cosmic string rainbow gravity spacetime
and a uniform magnetic field

It is convenient to discuss the KG-particles with a standard constant mass,
i.e., $f\left(r\right)=1\Longleftrightarrow$ $M\left(r\right)=0$, so that
Eq.(16) reduces into the two-dimensional Schrödinger-oscillator form

$\left\\{\partial_{r}^{2}-\frac{\left(\tilde{\ell}^{2}-1/4\right)}{r^{2}}+\frac{\tilde{\ell}\,\tilde{B}}{r}+\lambda\right\\}R\left(r\right)=0.$
(17)

Which obviously admits exact solution in the form of hypergeometric function
so that

$R\left(r\right)\sim\,\left(2i\sqrt{\lambda}r\right)^{|\tilde{\ell}|+1/2}\exp\left(-i\sqrt{\lambda}r\right)\,_{1}F_{1}\left(\frac{1}{2}+\tilde{\ell}|-\frac{\tilde{\ell}\,\tilde{B}}{2i\sqrt{\lambda}},1+2|\tilde{\ell}|,2i\sqrt{\lambda}r\right).$
(18)

However, to secure finiteness and square integrability we need to terminate
the hypergeometric function into a polynomial of degree $n_{r}\geq 0$ so that
the condition

$\frac{1}{2}+\tilde{\ell}|-\frac{\tilde{\ell}\,\tilde{B}}{2i\sqrt{\lambda}}=-n_{r}.$
(19)

is satisfied. This would in turn imply that

$i\sqrt{\lambda}=\frac{\tilde{\ell}\,\tilde{B}}{2\tilde{n}};\;\tilde{n}=n_{r}+|\tilde{\ell}|+\frac{1}{2}\Rightarrow\lambda_{n_{r},\ell}=-\frac{\tilde{\ell}\,^{2}\tilde{B}^{2}}{4\tilde{n}^{2}},$
(20)

and

$\psi\left(r\right)=\frac{R\left(r\right)}{\sqrt{r}}=\mathcal{N}\,r^{|\tilde{\ell}|}\exp\left(-\frac{|\tilde{\ell}\,\tilde{B}|}{2\tilde{n}}\,r\right)\,_{1}F_{1}\left(-n_{r},1+2|\tilde{\ell}|,\frac{|\tilde{\ell}\,\tilde{B}|}{\tilde{n}}r\right).$
(21)

Consequently, Eq.(15) would read

$g_{{}_{0}}\left(y\right)^{2}E^{2}-m^{2}=g_{{}_{1}}\left(y\right)^{2}\,K_{n_{r},\ell};\;K_{n_{r},\ell}=\frac{\tilde{B}^{2}}{4}\left[1-\frac{\tilde{\ell}^{2}}{\left(n_{r}+|\tilde{\ell}|+\frac{1}{2}\right)^{2}}\right]+k_{z}^{2}.$
(22)

At this point, it is convenient to mention that the choice of
$i\sqrt{\lambda}=\tilde{\ell}\,\tilde{B}/2\tilde{n}\geq 0\Rightarrow
i\sqrt{\lambda}=|\tilde{\ell}\,\tilde{B}|/2\tilde{n}$ is manifested by the
requirement of finiteness and square integrability of $\psi\left(r\right)$ as
$r\rightarrow\infty$. Interestingly, moreover, we notice that all $S$-states
(i.e., $\ell=0$ states) are degenerate with each other (positive with positive
and negative with negative states) and have the same $K_{n_{r},0}$ value

$K_{0,0}=K_{1,0}=\cdots=K_{n_{r},0}=\tilde{B}^{2}/4+k_{z}^{2}$ (23)

as suggested by Eq.(22). This is a consequence of the cosmic string spacetime
(i.e., at $\lim\limits_{\epsilon\rightarrow
0}g_{{}_{0}}\left(y\right)=\lim\limits_{\epsilon\rightarrow
0}g_{{}_{1}}\left(y\right)=1$) and has nothings to do with rainbow gravity.
Moreover, for a given $n_{r}$ and $\ell$ we have
$K_{n_{r},|\ell|}=K_{n_{r},-|\ell|}$, which indicates that we shall have
eminent degeneracies associated with the magnetic quantum number
$\ell=\pm|\ell|\neq 0$ for each radial quantum number $n_{r}$. These effects
are going to be reflected on the spectroscopic structure of constant mass KG-
particles in cosmic string rainbow gravity spacetime, under different sets of
rainbow functions.

To observe the rainbow gravity effects on such constant mass KG-particles we
now consider different rainbow functions.

###  III.1 The set of rainbow functions $g_{{}_{0}}\left(y\right)=1$,
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{n}}$

We start with the set $g_{{}_{0}}\left(y\right)=1$ and
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{2}}$ (i.e., $n=2$). Under such
rainbow functions structure the energy levels of (22) are given by

$E^{2}-m^{2}=\left(1-\epsilon\frac{E^{2}}{E_{p}^{2}}\right)K_{n_{r},\ell}\Longrightarrow
E_{n_{r},\ell}=\pm\sqrt{\frac{K_{n_{r},\ell}+m^{2}}{1+\delta\,K_{n_{r},\ell}}};\;\delta=\frac{\epsilon}{E_{p}^{2}}.$
(24)

We plot, in Figure 1(a), the corresponding energies against
$\delta=\epsilon/E_{p}^{2}$. We observe, for a given radial quantum number
$n_{r}$, eminent clustering of positive/negative energy levels as $\delta$
grows up from zero (i.e., the cosmic string spacetime limit). In Figure 1(b),
moreover, we plot the energies against $\left|eB_{\circ}\right|$. It is
obvious that as $\left|eB_{\circ}\right|\rightarrow 0$ the energy levels
converge to the values
$E_{n_{r},\ell}\sim\pm\sqrt{\left(k_{z}^{2}+m^{2}\right)/\left(1+\delta
k_{z}^{2}\right)}=\sqrt{2/1.1}$ for $\delta=0.1$, and $m=k_{z}=1$ value used
here. That is, at this limit positive/negative energy states emerge from the
same positive/negative values irrespective of the values of the radial and the
magnetic quantum numbers $n_{r}$ and $\ell$. On the other hand, as
$\left|eB_{\circ}\right|>>1$, the energy levels cluster and merge into
$E_{n_{r},\ell}\sim\pm\sqrt{1/\delta}$ (this is observed in Figure 1(b), i.e.,
as $\left|eB_{\circ}\right|>>1$ the energies
$E_{n_{r},\ell}\sim\pm\sqrt{1/\delta}=\pm\sqrt{10}$ for $\delta=0.1$). We may,
therefore, conclude that under such rainbow functions structure the energy
levels are destined to be within the range $\sqrt{(k_{z}^{2}+m^{2})/(1+\delta
k_{z}^{2})}\leq|E_{n_{r},\ell}|\,\leq\sqrt{1/\delta}=E_{p}/\sqrt{\epsilon}$.

Figure 1:  The energy levels of (24), using $\alpha=1/4$, $m=k_{z}=1$, so that
(a) shows $E$ against $\delta=\epsilon/E_{p}^{2}$ for $|eB_{\circ}|=1$,
$n_{r}=2$, $\ell=0,1,3,5,8$, and (b) shows $E$ against $|eB_{\circ}|$ for
$\delta=0.1$, $n_{r}=2$, $\ell=0,1,3,5,8$.

Figure 2:  The energy levels of (25), using $\alpha=1/4$, $m=k_{z}=1$, so that
(a) shows $E$ against $\beta=\epsilon/2E_{p}$ for $|eB_{\circ}|=1$, $n_{r}=2$,
$\ell=0,1,3,6$, and (b) shows $E$ against $|eB_{\circ}|$ for $\beta=0.1$,
$n_{r}=2$, $\ell=0,1,3,5,8$.

Now we consider the set $g_{{}_{0}}\left(y\right)=1$ and
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y}$ (i.e., $n=1$) in Eq.(22) to
obtain

$E^{2}-m^{2}=\left(1-\epsilon\frac{E}{E_{p}}\right)K_{n_{r},\ell}\Longrightarrow
E_{n_{r},\ell}=-\beta
K_{n_{r},\ell}\pm\sqrt{\beta^{2}K_{n_{r},\ell}^{2}+K_{n_{r},\ell}+m^{2}};\;\beta=\frac{\epsilon}{2E_{p}}.$
(25)

In Figures 2(a) and 2(b), we plot the energy levels against
$\beta=\epsilon/2E_{p}$ and $\left|eB_{\circ}\right|$, respectively. It is
obvious that the symmetry of the energy levels about $E=0$ is broken as an
effect of such rainbow functions structure. In Fig.2(a), we observe the
asymptotic tendency of the energy levels as $\beta\rightarrow 0$ (i.e., the
cosmic string spacetime limit) and as $\beta>>1$. It is obvious that
$\beta\rightarrow 0\Rightarrow E_{n_{r},\ell}=\pm\sqrt{K_{n_{r},\ell}+m^{2}}$,
whereas $\beta>>1\Rightarrow E_{n_{r},\ell}\rightarrow 0$ in the upper half
and $E_{n_{r},\ell}\sim-2\beta K_{n_{r},\ell}$ in the lower half of the energy
levels. Moreover, Figure 2(b) shows similar trend of asymptotic convergence as
an effect of the magnetic field for a given value of the rainbow gravity
parameter $\beta=0.7$.

Figure 3:  The energy levels of (26), using $\alpha=1/4$, $m=k_{z}=1$, so that
(a) shows $E$ against $\gamma=\epsilon m/E_{p}<1$ for $|eB_{\circ}|=1$,
$n_{r}=2$, $\ell=0,1,3,5,8$, and (b) shows $E$ against $|eB_{\circ}|$ for
$\gamma=0.5$, $n_{r}=2$, $\ell=0,1,3,5,8$.

Yet, in this case, it is more rapid since $K_{n_{r},\ell}\sim\tilde{B}^{2}$
for $\left|eB_{\circ}\right|>>1$ and consequently all energy levels cluster
around $E_{n_{r},\ell}\sim 1/2\beta$ ($=5$ for $\beta=0.1$ used in the figure)
in the upper half and $E_{n_{r},\ell}\sim-2a\tilde{B}^{2}$ (where
$0<a=\frac{1}{4}\left[1-\frac{\tilde{\ell}^{2}}{\left(n_{r}+|\tilde{\ell}|+\frac{1}{2}\right)^{2}}\right]<\frac{1}{4}$
) in the lower half. This effect is obvious from the energy levels in Eq.(25),
as the first negative term increases the negativity of the energy levels and
breaks the symmetry of the energy part of the second term.

###  III.2 The set of rainbow functions
$g_{{}_{0}}\left(y\right)=g_{{}_{1}}\left(y\right)=1/\left(1-\epsilon
y\right)$

Upon the substitution of
$g_{{}_{0}}\left(y\right)=g_{{}_{1}}\left(y\right)=1/\left(1-\epsilon
y\right)$ in Eq.(22) we obtain

$E^{2}-K_{n_{r},\ell}=\left(1-\epsilon\frac{E}{E_{p}}\right)^{2}m^{2}\Longrightarrow
E=\frac{-m\gamma\pm\sqrt{K_{n_{r},\ell}\left(1-\gamma^{2}\right)+m^{2}}}{1-\gamma^{2}};\;\gamma=\frac{\epsilon
m}{E_{p}}<1.$ (26)

In Figures 3(a) we plot the energy levels against $\gamma=\epsilon m/E_{p}<1$
to observe the rainbow gravity effect. We clearly see that the symmetry in the
energy levels is broken as an effect of the first term
$\left[-m\gamma/\left(1-\gamma^{2}\right)\right]$ in Eq.(26). In Figure 3(b)
the energy levels are plotted against $\left|eB_{\circ}\right|$ so that the
magnetic field effect on the energy levels is shown.

###  III.3 The set of rainbow functions $g_{{}_{0}}(y)=(e^{\epsilon
y}-1)/\epsilon y$ and $g_{{}_{1}}\left(y\right)=1$

We now use $g_{{}_{0}}(y)=(e^{\epsilon y}-1)/\epsilon y$ and
$g_{{}_{1}}\left(y\right)=1$ so that Eq.(22) implies

$E^{2}\left(\frac{e^{\epsilon E/E_{p}}-1}{\epsilon
E/E_{p}}\right)^{2}-m^{2}=K_{n_{r},\ell}\Longrightarrow
E=\frac{1}{2\beta}\ln\left(1\pm\sqrt{4\beta^{2}\left(K_{n_{r},\ell}+m^{2}\right)}\right);\;\beta=\frac{\epsilon}{2E_{p}}$
(27)

One should notice that $\left(\frac{e^{\epsilon E/E_{p}}-1}{\epsilon
E/E_{p}}\right)^{2}\rightarrow 1$ as $\epsilon\rightarrow 0$ (i.e., at the
cosmic string spacetime limit) and the energy levels retrieve their symmetry
about the $E=0$ line. In Figure 4(a) we plot the energy levels against
$\beta=\epsilon/2E_{p}$ and observe eminent clustering in the positive
energies as $\beta$ grows up from just above zero (i.e., $\beta\geq 0.001$),
whereas the negative energies are rapidly pushed further into the negative
energy region. In Figures 4(b) we show the effect of the magnetic field on the
energy levels.

Figure 4:  The energy levels of (27), using $\alpha=1/4$, $m=k_{z}=1$, so that
(a) shows $E$ against $\beta=\epsilon/2E_{p}$ for $|eB_{\circ}|=1$, $n_{r}=2$,
$\ell=0,1,2,4$, and (b) shows $E$ against $|eB_{\circ}|$ for $\beta=0.1$,
$n_{r}=2$, $\ell=0,1,2,3$.

##  IV PDM KG-particles in cosmic string rainbow gravity spacetime and a
uniform magnetic field

In this section, we consider a positive-valued dimensionless scalar multiplier
in the form of $f\left(r\right)=\exp\left(4\eta r\right)$ in Eq.(10) so that
$M(r)=\eta^{2}+\eta/r$. This would, in turn, imply that Eq.(16) reads

$\left\\{\partial_{r}^{2}-\frac{\left(\tilde{\ell}^{2}-1/4\right)}{r^{2}}+\frac{\left(-\eta+\tilde{\ell}\,\tilde{B}\right)}{r}+\tilde{\lambda}\right\\}R\left(r\right)=0,\;$
(28)

where

$\tilde{\lambda}=\frac{g_{{}_{0}}\left(y\right)^{2}E^{2}-m^{2}}{g_{{}_{1}}\left(y\right)^{2}}-\left(k_{z}^{2}+\frac{\tilde{B}^{2}}{4}\right)-\eta^{2},$
(29)

In this case, its exact solution is in the form of

$R(r)=C\,r^{|\tilde{\ell}|+1/2}\exp(-i\sqrt{\tilde{\lambda}}r)\,_{1}F_{1}\left(\frac{1}{2}+|\tilde{\ell}|-\frac{|-\eta+\tilde{\ell}\,\tilde{B}|}{2i\sqrt{\tilde{\lambda}}},1+2|\tilde{\ell}|,2i\sqrt{\tilde{\lambda}}r\right),$
(30)

which takes the form of a polynomial of degree $n_{r}\geq 0$ for

$\frac{1}{2}+|\tilde{\ell}|-\frac{|-\eta+\tilde{\ell}\,\tilde{B}|}{2i\sqrt{\tilde{\lambda}}}=-n_{r}\Rightarrow\frac{1}{2}+|\tilde{\ell}|-\frac{|-\eta+\tilde{\ell}\,\tilde{B}|}{2i\sqrt{\tilde{\lambda}}}=-n_{r}\Rightarrow
2i\sqrt{\tilde{\lambda}}=\frac{|-\eta+\tilde{\ell}\,\tilde{B}|}{\tilde{n}};\;\tilde{n}=n_{r}+|\tilde{\ell}|+\frac{1}{2}.$
(31)

Consequently, the eigen energies and wavefunctions, with
$\tilde{\eta}=-\eta+\tilde{\ell}\,\tilde{B}$, are given, respectively, by

$\tilde{\lambda}_{n_{r},\ell}=-\frac{\tilde{\eta}^{2}}{4\tilde{n}^{2}},$ (32)

and

$\psi\left(r\right)=\frac{R\left(r\right)}{\sqrt{r}}=C\,r^{|\tilde{\ell}|}\exp\left(-\frac{|\tilde{\eta}|}{2\tilde{n}}r\right)\,_{1}F_{1}\left(-n_{r},1+2|\tilde{\ell}|,\frac{|\tilde{\eta}|}{\tilde{n}}r\right).$
(33)

Hence, Eq.s (29) and (32) along with (15) would result

$g_{{}_{0}}\left(y\right)^{2}E^{2}-m^{2}=g_{{}_{1}}\left(y\right)^{2}\,\tilde{K}_{n_{r},\ell};\;\tilde{K}_{n_{r},\ell}=\left[k_{z}^{2}+\frac{\tilde{B}^{2}}{4}\left(1-\frac{\tilde{\ell}^{2}}{\tilde{n}^{2}}\right)+\eta^{2}\left(1-\frac{1}{4\tilde{n}^{2}}\right)+\frac{\eta\tilde{\ell}\,\tilde{B}}{2\tilde{n}^{2}}\right].$
(34)

Evidently, the last term of $\tilde{K}_{n_{r},\ell}$ in (34) would lift the
degeneracies associated with $\ell=\pm|\ell|\neq 0$ and states with both
$\ell$ values would reappear in the spectrum, therefore. One should, moreover,
notice that as $\tilde{B}\rightarrow 0$ our $\tilde{K}_{n_{r},\ell}\rightarrow
k_{z}^{2}+\eta^{2}\left(1-1/4\tilde{n}^{2}\right)$ and consequently states
with a specific $\ell=\pm|\ell|$ will emerge from the same $\tilde{B}=0$ and
split as $\tilde{B}$ grows up from zero. We may now discuss the effects of
rainbow gravity for different rainbow functions on the energy levels under the
current metaphoric PDM KG-particles in cosmic string spacetime and uniform
magnetic field of Eq.(34).

###  IV.1 The set of rainbow functions $g_{{}_{0}}\left(y\right)=1$,
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{n}}$

For rainbow functions $g_{{}_{0}}\left(y\right)=1$ and
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{2}}$ (i.e., $n=2$), Eq.(34)
would result

$E^{2}-m^{2}=\left(1-\epsilon\frac{E^{2}}{E_{p}^{2}}\right)\tilde{K}_{n_{r},\ell}\Longrightarrow
E=\pm\sqrt{\frac{\tilde{K}_{n_{r},\ell}+m^{2}}{1+\delta\,\tilde{K}_{n_{r},\ell}}};\;\delta=\frac{\epsilon}{E_{p}^{2}}.$
(35)

In Figures 5(a), we plot the energies against $\delta=\epsilon/E_{p}^{2}$. We
observe that both halves of the energy levels asymptotically converge to $E=0$
value as $\delta>>1$ for a fixed value of the PDM parameter $\eta$. In Fig.
5(b), we plot the energies against the PDM parameter $\eta$ and notice that
the degeneracies associated with $\ell=\pm|\ell|$ are removed as $\eta$
increases from zero (i.e., it is obvious that all states with
.$\ell=\pm|\ell|$ emerge form the same $\eta=0$ point and split as $\eta$
grows up). However, as $\eta>>1$ we observe that the energy levels merge into
$E_{n_{r},\ell}\sim\pm\sqrt{1/\delta}=\pm\sqrt{10}$ for $\delta=0.1.$In Figure
5(c), moreover, we plot the energies against $|eB_{\circ}|$ and observe that
the energy levels with a specific $\ell=\pm|\ell|$ split as $|eB_{\circ}|$
increases from the zero value, and for $|eB_{\circ}|>>1$ the energy levels
cluster and merge into $E_{n_{r},\ell}\sim\pm\sqrt{1/\delta}$ (this is
observed in Figure 5(c), i.e., as $|eB_{\circ}|>>1$ the energies
$E_{n_{r},\ell}\sim\pm\sqrt{1/\delta}=\pm\sqrt{10}$ for $\delta=0.1$). We
again observe that such rainbow functions structure the energy levels are
destined to be within the range $\sqrt{(k_{z}^{2}+m^{2})/(1+\delta
k_{z}^{2})}\leq|E_{n_{r},\ell}|\,\leq\sqrt{1/\delta}=E_{p}/\sqrt{\epsilon}$.

The second set of the rainbow functions $g_{{}_{0}}\left(y\right)=1$ and
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y}$ (i.e., $n=1$), on the other
hand, implies (using Eq. (34)) that

$E^{2}-m^{2}=\left(1-\epsilon\frac{E}{E_{p}}\right)\tilde{K}_{n_{r},\ell}\Longrightarrow
E=-\beta\,\tilde{K}_{n_{r},\ell}\pm\sqrt{\beta^{2}\tilde{K}_{n_{r},\ell}^{2}+\tilde{K}_{n_{r},\ell}+m^{2}};\;\beta=\frac{\epsilon}{2E_{p}}.$
(36)

In Figures 6(a) and (b), we plot the energy levels against
$\beta=\epsilon/2E_{p}$ and $|eB_{\circ}|$, respectively. It is obvious that
the symmetry of the energy levels about $E=0$ is broken as an effect of such
rainbow functions structure. In Fig.6(a) we notice that the clustering around
the $S$-state with $\ell=0$ only occurs for the upper half of the KG-energies
(i.e., positive energies), whereas in the lower half we observe that the
splitting in the energy levels increases as $\beta$ increases from zero. In
the lower half of Fig.6(b) we see that the energy levels separation increases
as the magnetic field strength increases from zero. This effect is obvious
from the form of the energy levels in Eq.(36), as the first negative term
increases the negativity of the energy levels and breaks the symmetry of the
energy part of the second term. In Fig.6(c) we show the effect of the PDM
settings on the energy levels where the symmetry of the energy levels about
$E=0$ is broken because of the rainbow functions structure.

Figure 5:  The energy levels of (35), using $m=k_{z}=1$, so that (a) shows $E$
against $\delta=\epsilon/E_{p}^{2}$ for $\alpha=1/4$, $|eB_{\circ}|=1$,
$\eta=0.5$, $n_{r}=1$, $\ell=0,\pm 1,\pm 2$, (b) shows $E$ against $\eta$ for
$\alpha=1/4$, $|eB_{\circ}|=1$, $\delta=0.1$, $n_{r}=1$, $\ell=0,\pm 1,\pm 2$,
and (c) shows $E$ against $|eB_{\circ}|$ for $\alpha=1/4$, $\eta=2$,
$\delta=0.1$, $n_{r}=1$, $\ell=0,\pm 1,\pm 2$.

Figure 6:  The energy levels of (36), using $\alpha=1/8$, $m=k_{z}=1$, so that
(a) shows $E$ against $\beta=\epsilon/2E_{p}$ for $|eB_{\circ}|=1$, $\eta=2$,
$n_{r}=1$, $\ell=0,\pm 1,\pm 2$, (b) shows $E$ against $|eB_{\circ}|$ for
$\eta=2$, $\beta=0.1$, $n_{r}=1$, $\ell=0,\pm 1,\pm 2$, and (c) shows $E$
against $\eta$ for $|eB_{\circ}|=1$, $\beta=0.1$, $n_{r}=1$, $\ell=0,\pm 1,\pm
2$.

###  IV.2 The set of rainbow functions
$g_{{}_{0}}\left(y\right)=g_{{}_{1}}\left(y\right)=1/\left(1-\epsilon
y\right)$

For the rainbow functions
$g_{{}_{0}}\left(y\right)=g_{{}_{1}}\left(y\right)=1/\left(1-\epsilon
y\right)$, Eq.(34) yields

$E^{2}-\tilde{K}_{n_{r},\ell}=\left(1-\epsilon\frac{E}{E_{p}}\right)^{2}m^{2}\Longrightarrow
E=\frac{-m\gamma\pm\sqrt{\tilde{K}_{n_{r},\ell}\left(1-\gamma^{2}\right)+m^{2}}}{1-\gamma^{2}};\;\gamma=\frac{\epsilon
m}{E_{p}}<1.$ (37)

In Figures 7(a) we plot the energy levels against $\gamma=\epsilon m/E_{p}$ to
observe the rainbow gravity effect. We clearly see that the symmetry in the
energy levels is broken as an effect of the first term
$\left[-m\gamma/\left(1-\gamma^{2}\right)\right]$ in Eq.(37). In Figures 7(b)
the energy levels are plotted against $|eB_{\circ}|$ so that the magnetic
field effect on the energy levels is shown, and in 7(c) we show the effect of
the PDM settings on the energy levels.

Figure 7:  The energy levels of (37), using $\alpha=1/8$, $m=k_{z}=1$, so that
(a) shows $E$ against $\gamma=\epsilon m/E_{p}<1$ for $|eB_{\circ}|=1$,
$\eta=2$, $n_{r}=1$, $\ell=0,\pm 1,\pm 2$, (b) shows $E$ against
$|eB_{\circ}|$ for $\eta=2$, $\gamma=0.1$, $n_{r}=1$, $\ell=0,\pm 1,\pm 2$,
and (c) shows $E$ against $\eta$ for $|eB_{\circ}|=1$, $\gamma=0.1$,
$n_{r}=1$, $\ell=0,\pm 1,\pm 2$.

Figure 8:  The energy levels of (38), using $\alpha=1/8$, $m=k_{z}=1$, so that
(a) shows $E$ against $\beta=\epsilon/2E_{p}$ for $|eB_{\circ}|=1$, $\eta=2$,
$n_{r}=1$, $\ell=0,\pm 1,\pm 2$, (b) shows $E$ against $|eB_{\circ}|$ for
$\eta=2$, $\beta=0.1$, $n_{r}=1$, $\ell=0,\pm 1,\pm 2$, and (c) shows $E$
against $\eta$ for $|eB_{\circ}|=1$, $\beta=0.1$, $n_{r}=1$, $\ell=0,\pm 1,\pm
2$.

###  IV.3 The set of rainbow functions $g_{{}_{0}}(y)=(e^{\epsilon
y}-1)/\epsilon y$ and $g_{{}_{1}}\left(y\right)=1$

Such rainbow functions structure in Eq.(34) would yield

$E^{2}\left(\frac{e^{\epsilon E/E_{p}}-1}{\epsilon
E/E_{p}}\right)^{2}-m^{2}=\tilde{K}_{n_{r},\ell}\Longrightarrow
E=\frac{1}{2\beta}\ln\left(1\pm\sqrt{4\beta^{2}\left(\tilde{K}_{n_{r},\ell}+m^{2}\right)}\right);\;\beta=\frac{\epsilon}{2E_{p}}$
(38)

In Figure 8(a) we plot the energy levels against $\beta=\epsilon/2E_{p}$ and
observe eminent clustering in the positive energies as $\beta$ grows up from
just above zero (i.e., $\beta\geq 0.001$), whereas the negative energies are
rapidly pushed further into the negative energy region. In Figures 8(b) we
show the effect of the magnetic field and in Fig. 8(c) we show the effect of
the PDM settings on the energy levels.

##  V Concluding remarks

We have considered KG-particles in the cosmic string rainbow gravity spacetime
(2) and uniform magnetic field (i.e.,
$\mathbf{B}=\mathbf{\nabla}\times\mathbf{A}=B_{\circ}\,\hat{z}$). We have
shown that the corresponding KG-equation reduces to the two-dimensional radial
Schrödinger Coulomb-like model (hence, the KG-Coulombic particle notion is
used in the process). The exact textbook solution of which is used (along with
the RG-modified energy-momentum dispersion relation (5)) and the effects of
rainbow gravity on the spectra are discussed. We have also explored the
effects of PDM (metaphorically speaking) on KG-Coulombic particles in cosmic
string rainbow gravity and uniform magnetic field. In the process, we have
studied the effects of four pairs of rainbow functions: (i)
$g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon
y^{2}}$, (ii) $g_{{}_{0}}\left(y\right)=1$,
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y}$, (iii)
$g_{{}_{0}}\left(y\right)=g_{{}_{1}}\left(y\right)=\left(1-\epsilon
y\right)^{-1}$, and (iv) $g_{{}_{0}}(y)=(e^{\epsilon y}-1)/\epsilon y$ and
$g_{{}_{1}}\left(y\right)=1$.

Among the four pairs of rainbow functions, only $[g_{{}_{0}}\left(y\right)=1$,
$g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon y^{2}}]$ provided energy levels
that are symmetric about $E=0$ line. Yet, it is interesting to observe that
for this particular pair of rainbow functions, the energy levels are destined
to be within the range $\sqrt{(k_{z}^{2}+m^{2})/(1+\delta
k_{z}^{2})}\leq|E_{n_{r},\ell}|\,\leq\sqrt{1/\delta}=E_{p}/\sqrt{\epsilon}$ as
the value of $|eB_{\circ}|$ increases from zero (i.e., zero charge $e=0$
and/or zero magnetic field strength, $B_{\circ}=0$). This effect is also
observed for PDM (using the same pair of rainbow functions). This effect is
documented in Figures 1(b) and 5(c). Evidently, moreover, for $\epsilon=1$ we
obtain the maximum possible value of the energy, $|E_{n_{r},\ell}|_{max}$ , of
the probe KG-Coulombic particle (in cosmic string rainbow gravity spacetime
and a uniform magnetic field) as the Planck energy $E_{p}$ so that
$\sqrt{(k_{z}^{2}+m^{2})/(1+k_{z}^{2}/E_{p}^{2})}\leq|E_{n_{r},\ell}|\,\leq\sqrt{1/\delta}=E_{p}$.

We have, however, observed that the magnetic field did not remove the
degeneracies of the energy levels associated with the magnetic quantum number
$\ell=\pm|\ell|$, but the introduction of PDM-settings (through the PDM
parameter $\eta$) has allowed the magnetic field to split $\ell=+|\ell|$ from
$\ell=-|\ell|$. Which is, in fact, a common feature for the four pairs of
rainbow functions we have considered. Yet, without the PDM parameter $\eta$,
we have noticed that all $S$-states (i.e., $\ell=0$ states) are degenerate
with each other (positive with positive and negative with negative states) and
have the same $K_{n_{r},0}$ value as shown in Eq. (23). However, when $\eta$
is brought into action, such degeneracy is removed (documented in (34) and
Figures 5, 6, 7, and 8).

The current study, in fact, supports and emphasises Bezerra’s et al. R81
statement that rainbow gravity is not just merely a mathematical re-scaling of
both time and spatial coordinates. Rainbow gravity has deeply affected the
spectroscopic structures for different rainbow function structures. The most
interesting effect of which is observed for the pair
$[g_{{}_{0}}\left(y\right)=1$, $g_{{}_{1}}\left(y\right)=\sqrt{1-\epsilon
y^{2}}]$, which, in turn, implied that the energy of the probe KG-
particle/antiparticle can not be more than the Planck’s energy $E_{p}$. This
result clearly suggests that the Planck energy $E_{p}$, in the rainbow gravity
model, is not only yet another invariant energy scale alongside the speed of
light, but also a maximum possible particle/antiparticle (here, KG-particles)
energy value (e.g., R44 ). More investigations should be carried out in this
direction, we believe. Finally, to the best of our knowledge, the current
methodical proposal did not appear elsewhere.

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Linear capacitors are those for which $v^{C}_{b}(t)=q_{b}/C$, where $C$ is the
capacitance of the element and is independent of the charge in the capacitor:

$v^{C}_{b}(t)=\frac{q_{b}}{C}=\frac{1}{C}\int_{-\infty}^{t}i^{C}_{b}\left(t^{\prime}\right)dt^{\prime}\,,\quad\to\quad
i^{C}_{b}(t)=C\frac{dv^{C}_{b}(t)}{dt}=C\ddot{\phi}\,,$ (68)

and hence the energy stored in them can be expressed as:

$E_{b}^{C}(t)=\frac{1}{2C}\left(q_{b}(t)-q_{\text{offset}}\right)^{2}\,.$ (69)

Inductive elements are those for which the current is only a function of the
flux:

$i_{b}(t)=g\left(\phi_{b}(t)\right)\,.$ (70)

Linear inductors are those for which $i^{L}_{b}(t)=\phi_{b}/L$, where $L$ is
the inductance of the element and is independent of the flux across the
inductor:

$i^{L}_{b}(t)=\frac{\phi_{b}}{L}=\frac{1}{L}\int_{-\infty}^{t}v^{L}_{b}\left(t^{\prime}\right)dt^{\prime}\quad\to\quad
v^{L}_{b}(t)=L\frac{di_{b}^{L}(t)}{dt}\,.$ (71)

and hence the energy stored in them can be expressed as:

$E_{b}^{L}(t)=\frac{1}{2L}\left(\phi_{b}(t)-\phi_{\text{offset}}\right)^{2}\,.$
(72)

So far this discussion has been classical with no regard for quantum effects.
Nevertheless, if we build the circuits with a superconducting material, such
as aluminum, and cool them in a dilution refrigerator to a temperature
$k_{B}T\ll\hbar\omega_{0}$, the thermal fluctuations become negligible in
comparison with the quantum fluctuations associated with the resonant
frequency of the circuit $\omega_{0}$. These circuits are usually cooled to a
temperature around $20mK$, ensuring that quantum fluctuations dominate for
circuits operating at microwave frequencies of the order of $1-20$ GHz. In the
Hamiltonian description of the circuit this condition means that we can
promote the classical variables to quantum operators

$\displaystyle\phi$ $\displaystyle\rightarrow\widehat{\phi}$ $\displaystyle q$
$\displaystyle\rightarrow\widehat{q}$ $\displaystyle H$
$\displaystyle\rightarrow\widehat{H}$

where the flux and charge operators of the nodes must satisfy the commutation
relation

$\left[\phi_{i},q_{j}\right]=i\hbar\delta_{ij}\,.$

The only problem of our discussion so far is that with the linear capacitors
and inductors that we have introduced the type of circuits that one can build
are, for instance, LC circuits. These circuits behave as quantum harmonic
oscillators, where the average value of the flux (position) and charge
(momentum) operators follow the classical equations of motion. Quantum
mechanics is only revealed when one considers higher moments such as
$\braket{\phi^{2}}$ or $\braket{q^{2}}$, which have the problem of being
considerably harder to measure than the averages. Another problem of LC
circuits is that they present a harmonic spectrum, and hence are useless if
one wants to build a qubit with a clearly differentiable two level energy
subspace.

The key to build superconducting circuits with directly observable macroscopic
quantum effects and anharmonic spectrums is to use non-linear components. The
most commonly used non-linear and non-dissipative electrical component is the
Josephson junction. It consists of a “sandwich” of two superconductors
separated by an insulator, usually two aluminum electrodes separated by a 1nm-
thin oxide (alumina) layer. This component is modeled as a non-linear ideal
inductor in parallel with a capacitor, which accounts for the capacitor formed
by the parallel plates of the superconducting electrodes. The non-linear
Josephson inductance is characterized by the current-flux relation:

$i_{b}^{J}(t)=I_{0}\sin\left(\varphi_{b}(t)\right)$ (73)

where $\varphi_{b}(t)$ is the phase difference between the macroscopic
wavefunction at each side of the inductance. Here the macroscopic wavefunction
is that of the Bose-Einsten condensate formed by the Cooper pairs in the
superconductor. One can show [42, 43] that this phase is related to the
electric flux

$\partial_{t}\varphi_{b}(\mathbf{x},t)=\frac{2\pi}{\Phi_{0}}\partial_{t}\phi_{b}(\mathrm{x},t)\,,$
(74)

where $\Phi_{0}$ is the magnetic flux quantum

$\Phi_{0}=\frac{h}{2e}\,.$ (75)

Note that the presence of $2e$ instead of $e$ in the equation is because the
charge unit in a superconductor is the charge of a Cooper pair, fact that was
discovered in the first experimental measure of the magnetic flux quantum
[44]. The flux-phase relation allows us to write the current across a
Josephson junction as

$i_{b}^{J}(t)=I_{c}\sin\left(\frac{2\pi}{\Phi_{0}}\phi_{b}(t)\right)$ (76)

where $I_{c}$ is the Josephson junction’s critical current, usually in the
range of $I_{c}\sim 400-600$ nA, current above which the model presented here
breaks down. Notice how we can obtain from this equation the clearly non-
linear inductance of the Josephson junction

$L_{J}=\frac{v_{b}^{J}(t)}{di_{b}^{J}(t)/dt}=\frac{\Phi_{0}}{2\pi}\frac{1}{I_{c}\cos\left(2\pi\phi_{b}(t)/\Phi_{0}\right)}$
(77)

and the energy stored in a Josephson junction

$E_{b}^{J}(t)=E_{J}\left[1-\cos\left(\frac{\phi_{b}(t)-\phi_{\text{offset}}}{\Phi_{0}/2\pi}\right)\right]$
(78)

where $E_{J}$ is the Josephson energy and is related to its critical current
by $E_{J}=I_{c}\Phi_{0}/2\pi$. Note that this energy has a constant term that
we will ignore because it only produces an undetectable phase shift in the
wavefunctions.

Another condition of the flux-phase relation is called the fluxoid
quantization, and states that the total magnetic flux thorugh a
superconducting loop must be an integer number of the magnetic flux quantum.
The total magnetic flux through a loop is usually decomposed as the flux
through the branch elements of the loop plus any external flux, and hence the
fluxoid quantization can be written as:

$\oint_{C}\nabla\phi\cdot dl+\Phi_{\text{ext}}=n\Phi_{0}$ (79)

##  Appendix B Hamiltonian of a superconducting circuit

Following the procedure presented in [45] for circuit quantization, we find
that the general Lagrangian for the circuits contemplated in this document
reads:

$\mathcal{L}=\mathcal{L}_{0}+\mathcal{L}_{int}$ (1)

where
$\mathcal{L}_{0}=\mathcal{L}_{1}(\bm{\phi},\dot{\bm{\phi}})+\mathcal{L}_{2}(\bm{\phi}^{\prime},\dot{\bm{\phi}^{\prime}})$
is the sum of the Lagrangians of the single flux qubits [2], fig. 2, which ca
be derived from the energy stored in the components that we derived in the
previous chapter:

$\displaystyle\mathcal{L}_{q}(\bm{\phi},\dot{\bm{\phi}})=$
$\displaystyle\frac{1}{2}\dot{\bm{\phi}}\textbf{C}_{q}\dot{\bm{\phi}}+E_{J}\cos{\left(\frac{\phi_{1}-\phi_{0}}{\varphi_{0}}\right)}+$
(2)
$\displaystyle+E_{J}\cos{\left(\frac{\phi_{2}-\phi_{0}}{\varphi_{0}}\right)}+\alpha
E_{J}\cos{\left(\frac{\phi_{2}-\phi_{1}-\Phi}{\varphi_{0}}\right)}$

with $E_{J}$ the characteristic Josephson energy of the junctions, $\Phi$ the
externally induced flux, $\phi_{i}$ the flux variables in the nodes of the
circuit, and $\textbf{C}_{q}$ the capacitance matrix of the flux qubit which
depends on the characteristic capacitance of the junctions $C$ and the qubits
parameter $\alpha$. $\mathcal{L}_{int}$ gives the interaction between qubits
and depends on the specific coupling. For a Josephson junction connecting two
nodes, $i$ and $j^{\prime}$, of different flux qubits we obtain a contribution
of the form:

$\mathcal{L}_{int}^{JJ}=\gamma_{ij^{\prime}}^{JJ}E_{J}\cos{\left(\frac{\phi^{\prime}_{j}-\phi_{i}}{\varphi_{0}}\right)}$
(3)

While a capacitor coupling gives a contribution:

$\mathcal{L}_{int}^{cap}=\gamma_{ij^{\prime}}^{cap}\frac{C}{2}(\dot{\phi^{\prime}_{j}}-\dot{\phi_{i}})^{2}$
(4)

In both cases, $\gamma_{ij^{\prime}}$ represents the proportionality constant,
between coupling Josephson energy (capacitance) and the flux qubits reference
Josephson energy, $E_{J}$ (capacitance, $C$). Using canonical variables,
$Q_{i}=\frac{\partial\mathcal{L}}{\partial\dot{\phi}_{i}}$, and the Legendre
transformation,
$H(\mathbf{Q},\bm{\phi})=\mathbf{Q}\dot{\bm{\phi}}-\mathcal{L}(\bm{\phi},\dot{\bm{\phi}})$,
we can conclude from the previous Lagrangian (1) that the general Hamiltonian
for two coupled 3JJQ is:

$H=H_{0}+H_{\text{int}}$ (5)

Here,
$H_{0}=H_{1}(\mathbf{Q},\bm{\phi})+H_{2}(\mathbf{Q}^{\prime},\bm{\phi}^{\prime})$
is the sum of the single flux qubits Hamiltonians (c-shunted or not) whose
inverse capacitance matrix may be modified by the action of the coupling
(renormalization or capacitive loading [1]),
$\widetilde{\mathbf{C}}_{q}^{-1}$,

$\displaystyle
H_{q}(\mathbf{Q},\bm{\phi})=\frac{1}{2}\mathbf{Q}\widetilde{\mathbf{C}}_{q}^{-1}\mathbf{Q}-E_{J}\cos{\left(\frac{\phi_{1}-\phi_{0}}{\varphi_{0}}\right)}$
(6) $\displaystyle-
E_{J}\cos{\left(\frac{\phi_{2}-\phi_{0}}{\varphi_{0}}\right)}-\alpha
E_{J}\cos{\left(\frac{\phi_{2}-\phi_{1}-\Phi}{\varphi_{0}}\right)}$

Here ${H}_{int}$ describes the interaction between pairs of different flux
qubits. For the inductive coupling mediated by a Josephson junction, we find
that

$H_{\text{int}}^{JJ}=-\gamma^{JJ}_{ij^{\prime}}E_{J}\cos{\left(\frac{\phi^{\prime}_{j}-\phi_{i}}{\varphi_{0}}\right)}$
(7)

where only the two connected nodes, $i$ and $j$, are implicated and no
renormalization for the qubits Hamiltonians has to be considered. However, for
the electrostatic interaction mediated by a capacitor we get an interaction
term of the form

$H_{\text{int}}^{cap}=\mathbf{Q}\mathbf{C}_{c}^{-1}\mathbf{Q}^{\prime}$ (8)

which gives connections between all node charges of different qubits and
depends on the inverse mutual capacitance matrix, $\mathbf{C}_{c}^{-1}$. The
single qubits Hamiltonians are renormalized by rescaling the inverse of the
capacitance matrix,
$\mathbf{C}_{q}^{-1}(\alpha)\rightarrow\widetilde{\mathbf{C}}_{q}^{-1}(\alpha,\gamma^{cap}_{ij})$.
To fully understand this procedure it is necessary to define the capacitance
matrices and inverse capacitance matrices that have been mentioned during the
explanation. The full capacitance matrix for our system (including all nodes
in both qubits) is defined as follows

$\mathbf{C}=\begin{pmatrix}\widetilde{\mathbf{C}}_{q}&-\mathbf{C}_{c}\\\
-\mathbf{C}^{T}_{c}&\widetilde{\mathbf{C}}_{q}^{\prime}\end{pmatrix}.$ (9)

For example, the full capacitance matrix for the circuit in Fig. 21(a) is

${\mathbf{C}}=C\begin{pmatrix}1+\alpha+\beta&-(\alpha+\beta)&0&0\\\
-(\alpha+\beta)&1+\alpha+\beta+\gamma&-\gamma&0\\\
0&-\gamma&1+\alpha+\beta+\gamma&-(\alpha+\beta)\\\
0&0&-(\alpha+\beta)&1+\alpha+\beta\end{pmatrix}.$ (10)

Thus, $\widetilde{\mathbf{C}}_{q}$ is the renormalized (or not) 3JJQs
capacitance matrix whose elements have the form

$\displaystyle(\widetilde{\mathbf{C}}_{q})_{ii}=(\mathbf{C}_{q})_{ii}+\sum_{j^{\prime}}\gamma_{ij^{\prime}}C$
(11) $\displaystyle(\widetilde{\mathbf{C}}_{q})_{ij}=(\mathbf{C}_{q})_{ij}$
$\displaystyle(\widetilde{\mathbf{C}}_{q}^{\prime})_{i^{\prime}i^{\prime}}=(\mathbf{C}_{q})_{i^{\prime}i^{\prime}}+\sum_{j}\gamma_{ji^{\prime}}C$
$\displaystyle(\widetilde{\mathbf{C}}_{q}^{\prime})_{i^{\prime}j^{\prime}}=(\mathbf{C}_{q})_{i^{\prime}j^{\prime}}$

Here $\mathbf{C}_{q}=\mathbf{C}_{q}^{\prime}$ is the non-renormalized single
3JJQ capacitance matrix. The elements $(C_{q})_{ii}$ are given by the sum of
all the capacitances connected to the node $i$ when uncoupled, and the
elements $-(C_{q})_{ij}$ are given by the sum of all capacitances connecting
nodes $i$ and $j$.The $\gamma_{ij^{\prime}}$ are the coupling parameters in
(4) which are $0$ if there’s no capacitive coupling involving the
corresponding nodes. And $\mathbf{C}_{c}$ is the mutual capacitance which
accounts for the capacitive couplings between nodes

$(\mathbf{C}_{c})_{ij^{\prime}}=\gamma_{ij^{\prime}}C.$ (12)

This way, we find the renormalized inverse capacitance matrix of the qubits,
$\widetilde{\mathbf{C}}_{q}^{-1}$ and
$\widetilde{\mathbf{C}}_{q}^{{}^{\prime}-1}$, and the inverse mutual
capacitance matrix, $\mathbf{C}_{c}^{-1}$, by performing the inversion of the
full capacitance matrix of the system:

$\mathbf{C}^{-1}=\begin{pmatrix}\widetilde{\mathbf{C}}_{q}^{-1}&\mathbf{C}_{c}^{-1}\\\
(\mathbf{C}_{c}^{-1})^{T}&\widetilde{\mathbf{C}}_{q}^{{}^{\prime}-1}\end{pmatrix}.$
(13)

##  Appendix C Numerical method to solve a superconducting circuit

We have written the Hamiltonians of our circuits in terms of the charge/number
and flux/phase operators, and we have represented the wavefunctions of the
circuits in a base of eigenstates of the flux/phase operators. This can be
convenient for instance when one works with the harmonic approximation,
however, if we could represent the Hamiltonians of the circuits in the
discrete base of eigenstates of the number operator we could write the
Hamiltonian as an infinite matrix, which can be truncated to a cutoff, and use
any of the available eigensolvers to find the eigenstates and energies of the
circuit.

One can show [46] the canonical commutation relations between $\hat{\phi}$ and
$\hat{q}$ can be manipulated to find that

$e^{i\hat{\phi}/\varphi_{0}}\hat{q}=(\hat{q}-2e)e^{i\hat{\phi}/\varphi_{0}}\,.$
(14)

This means that the exponential of the phase operator is the generator of
displacements in the space of charges. Since the charge operator can be easily
written in the number basis

$\hat{q}=-2e\hat{n}=\sum_{n}(-2en)\ket{n}\\!\bra{n}\,,\quad n\in\mathbb{Z}\,,$
(15)

we can conclude that the effect of the exponential of the phase operator is
$e^{i\hat{\varphi}}\ket{n}=\ket{n-1}$, i.e. the exponential of the phase
operator is a ladder operator in the number base:

$e^{i\hat{\varphi}}=\sum_{n}|n-1\rangle\langle n|$ (16)

The energy stored in a Josephson depends on $\cos(\hat{\varphi})$, which can
be decomposed in terms of exponentials and written in the number base as

$\cos(\hat{\varphi})=\frac{1}{2}\sum_{n}\ket{n+1}\\!\bra{n}+\frac{1}{2}\ket{n}\\!\bra{n+1}\,,$
(17)

which means that the Josephson junction allows processes in which a Cooper
pair tunnels in or out of one of its electrodes. With this we have all of the
necessary ingredients to write the Hamiltonian of a 3JJQ.

##  Appendix D 2D Harmonic approximation of the 3JJQ

If we connect a capacitor of capacitance $\gamma C$ to the node 2 of the 3JJ
Flux qubit show in fig. 2, the Hamiltonian of the circuit at the degeneration
point becomes

$\hat{H}=\frac{1}{rd}\left[(\gamma+2)n^{2}_{+}+(4\alpha+2+\gamma)n^{2}_{-}+2\gamma
n_{+}n_{-}\right]-\big{[}2\cos\left(\varphi_{+}\right)\cos\left(\varphi_{-}\right)-\alpha\cos\left(2\varphi_{+}\right)\big{]}\,,$
(18)

where $d=|C|=2\alpha+1+\gamma(\alpha+1)$ is the determinant of the new
capacitance matrix. This Hamiltonian can be written in matrix form for any of
the potential minima if we introduce the Harmonic approximation, i.e.
expanding the potential in powers of $\varphi_{+}$ and $\varphi_{-}$ and
keeping the quadratic terms. The approximate Hamiltonian for any minimum
located at
$(\varphi_{+}\\!=\\!\varphi_{+}^{*},\,\varphi_{-}\\!=\\!\varphi_{-}^{*})$ is:

$\hat{H}^{h}=\frac{1}{2}\mathbf{n}^{\text{T}}\mathbf{Tn}+\frac{1}{2}\mathbf{\bm{\varphi}}^{\text{T}}\mathbf{V\bm{\varphi}}\,,\vspace{-5mm}$
(19)

where,

$\mathbf{n}=\left[\begin{array}[]{c}n_{+}\\\ n_{-}\\\
\end{array}\right],\quad\bm{\varphi}=\left[\begin{array}[]{c}\varphi_{+}-\varphi_{+}^{*}\\\
\varphi_{-}-\varphi_{-}^{*}\\\
\end{array}\right],\quad\mathbf{T}=\frac{2}{rd}\left[\begin{array}[]{cc}\gamma+2&\gamma\\\
\gamma&4\alpha+2+\gamma\\\
\end{array}\right],\quad\mathbf{V}=\left[\begin{array}[]{cc}\frac{4\alpha^{2}-1}{\alpha}&0\\\
0&\frac{1}{\alpha}\\\ \end{array}\right]\,.$ (20)

To solve this Hamiltonian we have to find a set of transformations that allow
us to write the kinetic and potential energies in diagonal form. Since the
potential energy is already diagonal our first step will be to use a scale
transformation to define new phase variables $\bm{\varphi}_{1}$ that absorb
the matrix V and allow us to write the potential energy as
$\frac{1}{2}\bm{\varphi}_{1}^{T}\bm{\varphi}_{1}$. Since V is a positive-
definite diagonal matrix we can always find its square root and use it as a
scale transformation. To ensure that our variables remain canonically
commuting we have to apply the inverse scale transformation to the number
variables. Applying these transformations:

$\bm{\varphi}_{1}=\mathbf{V}^{\frac{1}{2}}\bm{\varphi}\,,\quad\mathbf{n}_{1}=\mathbf{V}^{-\frac{1}{2}}\mathbf{n}\,,\quad\mathbf{T}_{1}=\mathbf{V}^{\frac{1}{2}}\mathbf{T}\mathbf{V}^{\frac{1}{2}}\,,\quad\to\quad\hat{H}^{h}=\frac{1}{2}\mathbf{n}_{1}^{\text{T}}\mathbf{T}_{1}\mathbf{n}_{1}+\frac{1}{2}\bm{\varphi}_{1}^{\text{T}}\bm{\varphi}_{1}\,.$

The second and final step is to apply a rotation to the number variables which
diagonalizes the kinetic energy matrix. Since a rotation is a unitary
transformations the potential energy will remain diagonal. We can define this
rotation as:

$\mathbf{R}=\left[\begin{array}[]{cc}\cos(\theta)&-\sin(\theta)\\\
\sin(\theta)&\cos(\theta)\\\
\end{array}\right]\,,\quad\tan(2\theta)=\frac{\mathbf{T}_{1}(1,2)+\mathbf{T}_{1}(2,1)}{\mathbf{T}_{1}(2,2)-\mathbf{T}_{1}(1,1)}\,.$
(21)

Applying this rotation we can finally write the Hamiltonian in diagonal form

$\widetilde{\bm{\varphi}}=\mathbf{R}\bm{\varphi}_{1}\,,\quad\widetilde{\mathbf{n}}=\mathbf{Rn}_{1}\,,\quad\widetilde{\mathbf{T}}=\mathbf{R}\mathbf{T}_{1}\mathbf{R}^{\text{T}}\,,\quad\to\quad\hat{H}^{h}=\frac{1}{2}\widetilde{\mathbf{n}}^{\text{T}}\widetilde{\mathbf{T}}\widetilde{\mathbf{n}}+\frac{1}{2}\widetilde{\bm{\varphi}}^{\text{T}}\widetilde{\bm{\varphi}}\,,$
(22)

and its ground state in the transformed phase variables:

$\braket{\widehat{\widetilde{\bm{\varphi}}}}{g}=g(\bm{\widetilde{\varphi}})=\left(\pi^{-2}|\widetilde{\mathbf{T}}|^{-\frac{1}{2}}\right)^{\frac{1}{4}}\exp{\left(-\frac{1}{2}\widetilde{\bm{\varphi}}^{\text{T}}\widetilde{\mathbf{T}}^{-\frac{1}{2}}\widetilde{\bm{\varphi}}\right)}\,.$

We can undo the transformations to rewrite this eigenstate as a function of
the original $\bm{\varphi}$ variables:

$\braket{\widehat{\bm{\varphi}}}{g}=g(\bm{\varphi})=\left(\pi^{-2}|\mathbf{A}|\right)^{\frac{1}{4}}\exp{\left(-\frac{1}{2}\bm{\varphi}^{\text{T}}\mathbf{A}\bm{\varphi}\right)}\,,\quad\mathbf{A}=\mathbf{V}^{\frac{1}{2}}\mathbf{R}^{\text{T}}\mathbf{T}_{2}^{-\frac{1}{2}}\mathbf{R}\mathbf{V}^{\frac{1}{2}}\,,\quad\mathbf{|A|}=\sqrt{\frac{|V|}{|T|}}\,.$
(23)

Fig. 26 shows these wavefunctions for different combinations of $\alpha$ and
$\gamma$. Finally, we can calculate the overlap between two current states at
different wells:

$\displaystyle\braket{g_{L}}{g_{R}}$
$\displaystyle=\frac{\sqrt{|\mathbf{A}|}}{\pi}\int
d^{2}\varphi\exp\left({-\frac{1}{2}\big{[}(\bm{\varphi}-\bm{\varphi}_{L})^{\text{T}}\mathbf{A}(\bm{\varphi}-\bm{\varphi}_{L})+(\bm{\varphi}-\bm{\varphi}_{R})^{\text{T}}\mathbf{A}(\bm{\varphi}-\bm{\varphi}_{R})\big{]}}\right)$
(24)
$\displaystyle=\frac{\sqrt{|\mathbf{A}|}}{\pi}\exp\left({-\frac{1}{2}\big{[}\bm{\varphi}_{L}^{\text{T}}\mathbf{A}\bm{\varphi}_{L}+\bm{\varphi}_{R}^{\text{T}}\mathbf{A}\bm{\varphi}_{R}\big{]}}\right)\int
d^{2}\varphi\exp\left({-\frac{1}{2}\bm{\varphi}^{\text{T}}2\mathbf{A}\bm{\varphi}+(\bm{\varphi}_{L}+\bm{\varphi}_{R})^{\text{T}}\mathbf{A}\bm{\varphi}}\right)$
$\displaystyle=\frac{\sqrt{|\mathbf{A}|}}{\pi}\exp\left({-\frac{1}{2}\big{[}\bm{\varphi}_{L}^{\text{T}}\mathbf{A}\bm{\varphi}_{L}+\bm{\varphi}_{R}^{\text{T}}\mathbf{A}\bm{\varphi}_{R}\big{]}}\right)\frac{2\pi}{\sqrt{|2\mathbf{A}|}}\exp\left({\frac{1}{4}(\bm{\varphi}_{L}+\bm{\varphi}_{R})^{\text{T}}\mathbf{A}(\bm{\varphi}_{L}+\bm{\varphi}_{R})}\right)$
$\displaystyle=\exp\left({-\frac{1}{4}(\bm{\varphi}_{L}-\bm{\varphi}_{R})^{\text{T}}\mathbf{A}(\bm{\varphi}_{L}-\bm{\varphi}_{R})}\right)$

Figure 26: Harmonic wavefunctions, eq. (23), for three potential minima as a
function of $\alpha$ and $\gamma$. The blue-filled contour plots show the
wavefunctions. Note in the colorbar that the contour levels have not been
distributed linearly along the height of the wavefunction. on top of a black
contour plot of the 3JJQ potential. The solid-black contour lines show the
3JJQ potential and are distributed linearly along the height of the potential.
The dashed-black contour lines also show the 3JJQ potential but are
distributed linearly along the height of the potential barrier through
$t_{1}$.