Magnetized Kerr-Taub-NUT spacetime and Kerr/CFT correspondence

Haryanto M<EMAIL_ADDRESS>
Center for Theoretical Physics,

Department of Physics, Parahyangan Catholic University,

Jalan Ciumbuleuit 94, Bandung 40141, Indonesia

We present a new solution in Einstein-Maxwell theory which can be considered
as the magnetized version of Kerr-Taub-NUT solution. Some properties of the
spacetime are discussed. We also compute the entropy of extremal black hole in
the spacetime by using the Kerr/CFT correspondence approach.

##  1 Introduction

Kerr spacetime is an exact solution to the vacuum Einstein equations
describing the spacetime outside a rotating mass [1, 2]. Adding the NUT
parameter to this solution yields the Kerr-Taub-NUT (KTN) spacetime which also
belongs to the vacuum Einstein system. Interestingly, the presence of NUT
parameter in the solution leads to a regular Kretschmann scalar at the origin,
in contrast to the generic Kerr black hole solution that possesses a ring
singularity inside the horizon. However, despite the non existence of ring
singularity hiding inside the horizon of KTN black hole, KTN spacetime suffers
the so-called conical singularity [2] which is related to the non $2\pi$
periodicity of angular coordinate $\phi$. In addition to this conic
singularity, another peculiar feature of a spacetime equipped with NUT
parameter is the closed timelike curve (CTC) [2] where a spacelike coordinate
with a periodicity can become timelike. Nonetheless, in spite of these obscure
properties of KTN spacetime, numerous studies have been performed to explore
its aspects [3, 4, 5, 6, 7, 8, 9]. Even a spacetime solution with NUT
parameter can also exist in theories beyond Einstein-Maxwell [10, 11, 12].

Adding electromagnetic field into the spacetime of Einstein-Maxwell theory
yield an electrovacuum system. One of the popular exact solutions to the
Einstein-Maxwell equations is the Kerr-Newman spacetime. This solution
describes spacetime outside a rotating charged and massive object. Moreover,
this solution can contain a black hole provided that the rotating collapsing
object can maintain some amount of electric charge during its gravitational
contraction [1]. Another way to add electromagnetic fields in the spacetime is
by magnetizing a solution following the Ersnt formalism [14]. Basically, this
Ersnt magnetization is a type of Harrison transformation acting on some known
seed solutions in Einstein-Maxwell theory. The review on Ersnt magnetization
can be found in literature [15, 16, 17], and one can view the Melvin magnetic
universe [18] can be obtained by magnetizing the Minkowski spacetime.

Particularly, applying the Ernst magnetization to the Kerr spacetime gives us
the magnetized Kerr solution [14] which is interpreted as the spacetime
outside a Kerr black hole immersed in a strong magnetic field. Unlike the
generic Kerr solution which has the asymptotic flatness, the magnetized
version is no longer flat at asymptotic due to the presence of homogeneous
external magnetic field filling in the spacetime. Indeed, this nature of
magnetized Kerr solution is found quite unrealistic to be associated to any
known astrophysical system, but one can still consider it as some approximate
way to model a black hole immersed in a strong magnetic field coming from the
accretion disc [16]. Therefore, it can be understood why this solution is
still worth further investigations, for example in its relation to the Meisner
effect [19] and the Kerr/CFT holography [20, 21]. Some other aspects of
magnetized black holes can be found in literature [17, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31]

The Kerr/CFT correspondence has been used as a tool to compute the entropy of
extremal Kerr black hole using some two dimensional conformal field theory
(CFT2) point of views [32, 33, 34]. The $SL\left(2,{\mathbb{R}}\right)\times
U\left(1\right)$ isometry of the near horizon geometry of the extremal black
hole is the window where the CFT2 method can apply. The obtained
diffeomorphisms subject to some boundary conditions can be brought to be a set
of Virasoro generators of a CFT2 whose commutator incorporates a specific
central charge. In turn, plugging the obtained central charge and Frolov-
Thorne temperature in Cardy formula, the entropy for extremal black hole can
be reproduced. This Kerr/CFT holography has been applied to many type of
rotating or charged black holes in the Einstein-Maxwell theory [34] and beyond
[35, 36].

Despite the real world application of the NUT parameter or external magnetic
field in the spacetime obeying the Einstien-Maxwell equations is still vague,
the discussions of spacetime exhibiting these properties have extended our
understanding of some aspects in gravitational theories [37, 38, 39, 40, 41,
42]. In particular in how we define the conserved quantities in the spacetime
such as mass and angular momentum, and also the geodesics of test particles in
the spacetime [6]. Also in a recent work [43], the authors show that the
Misner string contribution to the entropy of Taub-NUT-AdS can be renormalized
by introducing the Gauss-Bonnet term. Therefore, one may wonder what the
properties of spacetime with both NUT parameter and external magnetic field.
This is exactly the motivation in [44] where the author introduced the
magnetized version of the Taub-NUT solution, which was extended to the charged
case namely the Reissner–Nordstrom -Taub-NUT spacetime [45]. In this paper,
still in the same spirit, we would like apply the magnetization scheme to the
case of KTN spacetime to obtain the magnetized KTN (MKTN) solution which is to
the best of our knowledge does not exist in literature.

The organization of this paper is as follows. In the next section, we
construct the MKTN solution by employing the Ernst magnetization to the KTN
metric as the seed solution. Some aspects of the obtained spacetime such as
the surface area, squared of Riemann tensor at equator, and the Hawking
temperature are studied in section 3. In section 4, the near horizon geometry
of the black hole is obtained, followed by the microscopic entropy calculation
according to Kerr/CFT correspondence. Finally we give conclusions and
discussions. A brief review of Ernst magnetization is presented in the
appendix A. We consider the natural units $c={\hbar}=k_{B}=G_{4}=1$.

##  2 Magnetized Kerr-Taub-NUT spacetime

The MKTN spacetime presented in this section is a result of Ernst
magnetization to the KTN line element

$d{s^{2}}=-{{{\Delta_{r}}{{\left({dt-\left({a{\Delta_{x}}-2lx}\right)d\phi}\right)}^{2}}}\over\Xi}+{\rho^{2}}\left({{{d{r^{2}}}\over{{\Delta_{r}}}}+{{d{x^{2}}}\over{{\Delta_{x}}}}}\right)+{{{\Delta_{x}}}\over{\Xi}}{\left({adt-\left({{r^{2}}+{a^{2}}+{l^{2}}}\right)d\phi}\right)^{2}}$
(2.1)

as the seed. Above we have $\Xi={{{r^{2}}+{\left({l+ax}\right)^{2}}}}$,
${\Delta_{r}}={r^{2}}-2mr+{a^{2}}-{l^{2}}$, and ${\Delta_{x}}=1-{x^{2}}$. Note
that the KTN (2.1) solution above solves the vacuum Einstein equations, so
there is no electromagnetic field associated to the system. To apply the Ersnt
magnetization, first we need to rewrite the last metric into the Lewis-
Papapetrou-Weyl form (A.1), where the corresponding functions can be found as
the followings,

$f={{{\Delta_{x}}{{\left({{r^{2}}+{l^{2}}+{a^{2}}}\right)}^{2}}-{\Delta_{r}}{{\left({a{\Delta_{x}}-2lx}\right)}^{2}}}\over{{\Xi}}}\,,$
(2.2)
$\omega={{{\Delta_{r}}\left({a{\Delta_{x}}-2lx}\right)-a{\Delta_{x}}\left({{r^{2}}+{l^{2}}+{a^{2}}}\right)}\over{{\Delta_{r}}{{\left({a{\Delta_{x}}-2lx}\right)}^{2}}-{\Delta_{x}}{{\left({{r^{2}}+{l^{2}}+{a^{2}}}\right)}^{2}}}}\,,$
(2.3)
$e^{2\gamma}={{\Delta_{x}}{{\left({{r^{2}}+{l^{2}}+{a^{2}}}\right)}^{2}}-{\Delta_{r}}{{\left({a{\Delta_{x}}-2lx}\right)}^{2}}}\,,$
(2.4)

and $\rho^{2}=\Delta_{x}\Delta_{r}$. In the absence of electromagnetic field,
the existing Ernst potential is only $\cal E$, where the real and imaginary
components are given by

${{\mathop{\rm Re}\nolimits}{\cal
E}}=\frac{1}{\Xi}\left\\{\Delta_{x}r^{4}+\left({a}^{2}+4lax-6{l}^{2}{x}^{2}-{a}^{2}{x}^{4}+2{l}^{2}-4a{x}^{3}l\right)r^{2}+2m\left(2lx+a{x}^{2}-a\right)^{2}r\right.$
$\left.-\left(l+ax\right)\left({a}^{3}{x}^{3}-{a}^{3}x-a{x}^{3}{l}^{2}-3{l}^{3}{x}^{2}+3{a}^{2}{x}^{2}l+5{l}^{2}ax-3{a}^{2}l-{l}^{3}\right)\right\\}\,,$
(2.5)

and

${\mathop{\rm Im}\nolimits}{\cal
E}=\frac{1}{a^{2}\Xi}\left\\{2{a}^{2}l\left({x}^{2}+1\right)r^{3}-2m\left({a}^{3}{x}^{3}-3{a}^{3}x-2{a}^{2}l-{l}^{3}+3{a}^{2}{x}^{2}l\right)r^{2}+2{a}^{2}l\left(3{a}^{2}{x}^{2}\right.\right.$
$\left.\left.-2a{x}^{3}l+{l}^{2}+6lax-{a}^{2}-3{l}^{2}{x}^{2}\right)r+2m\left({a}^{2}+{l}^{2}\right)\left(l+ax\right)\left({a}^{2}{x}^{2}+{a}^{2}+lax+{l}^{2}\right)\right\\}\,,$
(2.6)

respectively. Following (A.9), the $\Lambda$ function associated to this Ernst
potential is given by

$\Lambda={{{{\tilde{\Lambda}}_{R}}+i{{\tilde{\Lambda}}_{I}}}\over{{a^{2}}\left({l+ax+ir}\right)}}\,,$
(2.7)

where

${{\tilde{\Lambda}}_{R}}=-{a}^{2}{b}^{2}\left(l+3l{x}^{2}-ax\Delta_{x}\right)r^{2}+2{b}^{2}m\left({a}^{3}{x}^{3}-3{a}^{3}x-{l}^{3}-2{a}^{2}l+3{a}^{2}l{x}^{2}\right)r$
$-a^{2}\left\\{\left(ab^{2}x^{3}+3b^{2}x^{2}l\right)\left(a^{2}-l^{2}\right)-ax\left(b^{2}{a}^{2}+1-5{b}^{2}{l}^{2}\right)-l\left(3{b}^{2}{a}^{2}+1+{b}^{2}{l}^{2}\right)\right\\}\,,$
(2.8)

and

${{\tilde{\Lambda}}_{I}}=b^{2}a^{2}\Delta_{x}r^{3}+\left\\{{b}^{2}\left(3\,{l}^{2}+{a}^{2}\right){x}^{2}-4a{b}^{2}lx-{b}^{2}{a}^{2}-3{b}^{2}{l}^{2}-1\right\\}r$
$2{b}^{2}m\left({l}^{2}+{a}^{2}\right)\left({a}^{2}{x}^{2}+{a}^{2}+lax+{l}^{2}\right)\,.$
(2.9)

Then the transformed Ernst potentials, which give us a set of magnetized
solution for the metric and vector potential, can be found using (A.8). The
resulting magnetized metric can be expressed as

${\rm{d}}s^{2}=\frac{1}{{f^{\prime}}}\left\\{{-\rho^{2}{\rm{d}}t^{2}+e^{2\gamma}\left({\frac{{{\rm{d}}r^{2}}}{{\Delta_{r}}}+\frac{{{\rm{d}}x^{2}}}{{\Delta_{x}}}}\right)}\right\\}+f^{\prime}\left({{\rm{d}}\phi-\omega^{\prime}{\rm{d}}t}\right)^{2}\,,$
(2.10)

where $f^{\prime}={\left|\Lambda\right|^{2}}f$, and

$\omega^{\prime}={{\sum\limits_{j=0}^{6}{{c_{j}}{x^{j}}}}\over{\sum\limits_{k=0}^{4}{{d_{k}}{x^{k}}}}}\,,$
(2.11)

with the corresponding $c_{j}$’s are given by

$c_{6}=2\left({l}^{2}{a}^{2}+3mr{a}^{2}-2{a}^{2}{m}^{2}+3{l}^{2}mr-{r}^{3}m-3{r}^{2}{l}^{2}-{l}^{4}-2{l}^{2}{m}^{2}\right){a}^{5}{b}^{4}\Delta_{r}\,,$
$c_{5}=-2\left(3{a}^{4}-6{l}^{2}{a}^{2}+8{a}^{2}{m}^{2}-6{r}^{2}{a}^{2}-8mr{a}^{2}-8{l}^{2}mr+3{l}^{4}-{r}^{4}+6{r}^{2}{l}^{2}+4{r}^{3}m+8{l}^{2}{m}^{2}\right){a}^{4}{b}^{4}l\Delta_{r}\,,$
$c_{4}=-2\left(12{l}^{4}{m}^{2}-15{l}^{4}{a}^{2}-12mr{l}^{4}+16{a}^{4}{l}^{2}-15mr{l}^{2}{a}^{2}-21{r}^{2}{l}^{2}{a}^{2}+14{m}^{2}{a}^{2}{l}^{2}\right.$
$\left.-7m{r}^{3}{a}^{2}-3mr{a}^{4}+4{m}^{2}{a}^{4}\right){a}^{3}{b}^{4}\Delta_{r}\,,$
$c_{3}=4\left({l}^{4}{a}^{2}+6mr{l}^{4}-4{l}^{4}{m}^{2}-20{a}^{4}{l}^{2}+8mr{l}^{2}{a}^{2}+6{r}^{2}{l}^{2}{a}^{2}-8{m}^{2}{a}^{2}{l}^{2}+2m{r}^{3}{l}^{2}+{a}^{6}+2mr{a}^{4}\right.$
$\left.+{a}^{2}{r}^{4}+6m{r}^{3}{a}^{2}-6{r}^{2}{a}^{4}-8{m}^{2}{a}^{4}\right){a}^{2}{b}^{4}l\Delta_{r}\,,$
$c_{2}=-3a^{4}mb^{4}r^{5}-6{a}^{4}{b}^{4}\left(3{l}^{2}-2{m}^{2}\right)r^{4}-2{a}^{2}{b}^{4}m\left(5{a}^{4}+4{l}^{4}-19{l}^{2}{a}^{2}\right)r^{3}-6{a}^{2}{b}^{4}\left({a}^{4}{l}^{2}-4{l}^{4}{m}^{2}-4{m}^{2}{a}^{4}\right.$
$\left.-6{m}^{2}{a}^{2}{l}^{2}\right)r^{2}-m\left\\{m^{2}4{b}^{4}\left({l}^{2}+3{a}^{2}\right)\left(-{l}^{4}-3{l}^{2}{a}^{2}+{a}^{4}\right){a}^{2}\left(7{a}^{6}{b}^{4}-{a}^{2}-69{a}^{2}{b}^{4}{l}^{4}+34{a}^{4}{b}^{4}{l}^{2}-24{b}^{4}{l}^{6}\right)\right\\}r\,,$
$c_{1}=2l\Delta_{r}\left\\{-3r^{4}a^{4}b^{4}-4{a}^{2}{b}^{4}m\left({l}^{2}+2{a}^{2}\right)r^{3}+6a^{4}b^{4}\left(a^{2}-l^{2}\right)r^{2}-12{a}^{2}{b}^{4}m\left({a}^{2}+{l}^{2}\right)^{2}r\right.$
$\left.+{a}^{8}{b}^{4}+{b}^{4}{a}^{4}{l}^{4}+18{a}^{6}{b}^{4}{l}^{2}+8{a}^{6}{b}^{4}{m}^{2}-12{a}^{4}{b}^{4}{m}^{2}{l}^{2}-{a}^{4}-4{b}^{4}{m}^{2}{l}^{6}-16{a}^{2}{b}^{4}{m}^{2}{l}^{4}\right\\}\,,$
$c_{0}=2a\left\\{-3{a}^{4}m{b}^{4}{r}^{5}-2{a}^{2}{b}^{4}m\left(2{l}^{4}+7{l}^{2}{a}^{2}+{a}^{4}\right)r^{3}+{a}^{2}{b}^{4}\left(3{a}^{4}{l}^{2}-8{m}^{2}{a}^{2}{l}^{2}-4{l}^{4}{m}^{2}-6{m}^{2}{a}^{4}\right)r^{2}\right.$
$+m\left({a}^{8}{b}^{4}-16{a}^{4}{b}^{4}{m}^{2}{l}^{2}-{a}^{4}-8{a}^{6}{b}^{4}{l}^{2}-27{b}^{4}{a}^{4}{l}^{4}-12{b}^{4}{l}^{6}{a}^{2}-4{b}^{4}{m}^{2}{l}^{6}-16{a}^{2}{b}^{4}{m}^{2}{l}^{4}\right)r$
$\left.-4{b}^{4}{m}^{2}{l}^{8}+4{a}^{2}{b}^{4}\left({a}^{2}-3{m}^{2}\right){l}^{6}+\left(13{a}^{6}{b}^{4}-4{a}^{4}{b}^{4}{m}^{2}\right){l}^{4}+\left(3{a}^{8}{b}^{4}+10{a}^{6}{b}^{4}{m}^{2}-{a}^{4}\right){l}^{2}+2{a}^{8}{b}^{4}{m}^{2}\right\\}\,,$

and the corresponding $d_{k}$’s are

$d_{4}=a^{6}\Delta_{r}\,,$ $d_{3}=4la^{5}\Delta_{r}\,,$
$d_{2}=-\left(8{l}^{2}mr+{a}^{4}-4mr{a}^{2}-{r}^{4}-8{l}^{2}{a}^{2}+3{l}^{4}-6{r}^{2}{l}^{2}\right){a}^{4}\,,$
$d_{1}=-4la^{5}\Delta_{r}\,,$
$d_{0}=-\left(2mr{a}^{2}+{r}^{2}{a}^{2}+3{l}^{2}{a}^{2}+{r}^{4}+{l}^{4}+2{r}^{2}{l}^{2}\right){a}^{4}\,.$

Note that the magnetized metric (2.10) possesses the Killing vectors
$\zeta^{\mu}_{\left(t\right)}\partial_{\mu}=\partial_{t}$ and
$\zeta^{\mu}_{\left(\phi\right)}\partial_{\mu}=\partial_{\phi}$ just like the
seed solution (2.1).

Unlike the seed solution which belongs to the vacuum Einstein, the magnetized
version is now contain electromagnetic field where the corresponding Ernst
potential is given by

$\Phi^{\prime}={{{{\cal F}_{R}}+i{{\cal F}_{I}}}\over{{{\cal G}_{R}}+i{{\cal
G}_{I}}}}\,,$ (2.12)

where

${\cal
F}_{R}=b\left\\{{a}^{2}\left(l-ax+a{x}^{3}+3l{x}^{2}\right)r^{2}-2\,m\left(-{l}^{3}+{a}^{3}{x}^{3}-3{a}^{3}x-2{a}^{2}l+3{a}^{2}l{x}^{2}\right)r\right.$
$\left.+{a}^{2}\left({a}^{3}{x}^{3}-{l}^{3}-a{l}^{2}{x}^{3}+5a{l}^{2}x-3{l}^{3}{x}^{2}-{a}^{3}x+3{a}^{2}l{x}^{2}-3{a}^{2}l\right)\right\\}\,,$
(2.13) ${\cal
F}_{I}=-b\left\\{a^{2}\Delta_{x}r^{2}-{a}^{2}\left(3{l}^{2}{x}^{2}-{a}^{2}\Delta_{x}-3{l}^{2}-4lax\right)r+m\left({a}^{2}+{l}^{2}\right)\left({a}^{2}{x}^{2}+{a}^{2}+lax+{l}^{2}\right)\right\\}\,,$
(2.14) ${\cal
G}_{R}=-{a}^{2}{b}^{2}\left(l-ax\Delta_{x}+3l{x}^{2}\right)r^{2}+2{b}^{2}m\left(3{a}^{2}l{x}^{2}-{l}^{3}+{a}^{3}{x}^{3}-3{a}^{3}x-2{a}^{2}l\right)r-{a}^{2}\left({b}^{2}{a}^{3}{x}^{3}\right.$
$\left.-{l}^{3}{b}^{2}-{b}^{2}{a}^{3}x-3{b}^{2}{a}^{2}l-3{x}^{2}{l}^{3}{b}^{2}-l-{b}^{2}a{l}^{2}{x}^{3}+5{b}^{2}a{l}^{2}x+3{b}^{2}{a}^{2}{x}^{2}l-ax\right)\,,$
(2.15) ${\cal
G}_{I}=b^{2}a^{2}\Delta_{x}r^{2}-{a}^{2}\left(3{x}^{2}{b}^{2}{l}^{2}-4a{b}^{2}lx-3{l}^{2}{b}^{2}-{b}^{2}{a}^{2}-1+{b}^{2}{a}^{2}{x}^{2}\right)r$
$+2{b}^{2}m\left({a}^{2}+{l}^{2}\right)\left({a}^{2}{x}^{2}+{a}^{2}+lax+{l}^{2}\right)\,.$
(2.16)

From this potential, the components of vector field $A_{\mu}$ can be obtained
using eq. (A.2), (A.13), and (A.14) accordingly. However, the full expression
of $A_{\phi}$ and ${\tilde{A}}_{\phi}$ are tedious, and we provide them in
appendix B. Furthermore, getting the full expression for $A_{t}$ is
troublesome since we need to integrate $A_{\phi}$ and ${\tilde{A}}_{\phi}$.
Therefore, since in this present work we do not need the full mathematical
expression for $A_{\mu}$, we will omit to present the complete result for
$A_{t}$. In using some symbolic manipulation programs we can show the obtained
solution solves the Einstein-Maxwell equations by using the field strength
tensor $F_{\mu\nu}$ constructed from the $A_{\phi}$ solution in (B.1) and
$\partial_{r}A_{t}$ together with $\partial_{x}A_{t}$ obeying (A.13) and
(A.14), respectively.

##  3 Some aspects of the black hole

The spacetime (2.10) introduced in this paper can contain a black hole with
radius $r_{+}=m+\sqrt{m^{2}-a^{2}+l^{2}}$, exactly the same as the radius of
KTN horizon. The total area of the black hole then can be computed using the
standard textbook formula

${\cal
A}=\int\limits_{0}^{2\pi}{d\phi\int\limits_{-1}^{1}{dx\sqrt{g_{\phi\phi}g_{xx}}}}=4\pi\left(r_{+}^{2}+a^{2}+l^{2}\right)\,,$
(3.1)

which is equal to the area of KTN black hole [8]. Following the relation where
the entropy of a black hole is a quarter of its area [8], the entropy of a
MKTN black hole reads

$S=\frac{\cal A}{4}=\pi\left(r_{+}^{2}+a^{2}+l^{2}\right)\,.$ (3.2)

To grasp more aspects of this new spacetime, let us here examine the squared
of corresponding Riemann curvature tensor. Surely expressing this quantity in
its full complicated and lengthy expression, even on equator, is unnecessary.
Then we would rather to present some of its numerical evaluations as presented
in fig. 3.1. The typical curvature in the spacetime with NUT parameter is
obvious in the red and black lines, where they intersect the vertical axis of
$r=0$ at some finite values of
$K^{*}=m^{-4}R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}$. We can compare to
the blue line which represents the curvature in generic Kerr spacetime, where
there exist singularity at the origin. This confirms our earlier statement
that rotating spacetimes with NUT parameter, including MKTN, does not possess
the true ring singularity as the null NUT parameter counterpart exhibits.

Figure 3.1: Plots of dimensionless squared Riemann tensor
$K^{*}=m^{-4}R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}$ at $x=0$ for the
spacetime (2.10) where we have considered the numerical value $a=0.2~{}m$.

Now let us turn to a semiclassical aspect where we would like to compute the
Hawking temperature associated to a black hole in MKTN spacetime. Here we
adopt the complex path method proposed in [46] which has been applied to a
Taub-NUT black hole in [47]. We start with a general form of metric

$ds^{2}=-\tilde{f}\left({r,x}\right)dt^{2}+\frac{{dr^{2}}}{{\tilde{g}\left({r,x}\right)}}+\tilde{C}\left({r,x}\right)h_{ij}\left({r,x}\right)d\tilde{x}^{i}d\tilde{x}^{j}\,,$
(3.3)

where $\tilde{x}^{i}=\left[{x,\tilde{\phi}}\right]$, and
${\tilde{\phi}}=\phi-\omega^{\prime}t$. Obviously the metric (2.10) can be
rewritten in the form (3.3) above, hence the complex path approach for Hawking
radiation presented in [47] can apply. Now we consider the geodesic
$dx=d{\tilde{\phi}}=0$, for a fixed $x=0$, then the $\left(t-r\right)$ sector
of metric above which matters. In such consideration and using the Hamilton-
Jacobi ansatz for the scalar field
$\Phi=\exp\left[-iS\left(t,r\right)\right]$, the reading of massless Klein-
Gordon equation $\nabla_{\mu}\nabla^{\mu}\Phi$ can be written in the form

$\left({\frac{{\partial S}}{{\partial
r}}}\right)^{2}=\frac{1}{{\tilde{f}\left(r\right)\tilde{g}\left(r\right)}}\left({\frac{{\partial
S}}{{\partial t}}}\right)^{2}\,.$ (3.4)

As the spacetime being stationary, one can look for a solution

$S\left({t,r}\right)=Et+\tilde{S}\left(r\right)\,.$ (3.5)

Then the solution to eq. (3.4) can be written as

$S\left({t,r}\right)=E\left({t\pm\int\limits_{0}^{r}{\frac{{dr}}{{\sqrt{\tilde{f}\left(r\right)\tilde{g}\left(r\right)}}}}}\right)\,.$
(3.6)

Plugging the last result into our Hamilton-Jacobi ansatz yields the ingoing
and outgoing modes

$\Phi_{{\rm{in}}}=\exp\left[{-iE\left({t+\int\limits_{0}^{r}{\frac{{dr}}{{\sqrt{\tilde{f}\left(r\right)\tilde{g}\left(r\right)}}}}}\right)}\right]\,,$
(3.7)

and

$\Phi_{{\rm{out}}}=\exp\left[{-iE\left({t-\int\limits_{0}^{r}{\frac{{dr}}{{\sqrt{\tilde{f}\left(r\right)\tilde{g}\left(r\right)}}}}}\right)}\right]\,,$
(3.8)

respectively.

Obviously, the probability of ingoing particle is unity, i.e. $P_{{\mathop{\rm
in}\nolimits}}=\left|{\Phi_{{\rm{in}}}}\right|^{2}=1$. Then by using the
detailed balance principle

$\frac{{P_{{\rm{out}}}}}{{P_{{\rm{in}}}}}=\exp\left[{-\frac{E}{{T_{H}}}}\right]\,,$
(3.9)

one can write the Hawking temperature

$T_{H}=\frac{1}{4}\left({{\mathop{\rm
Im}\nolimits}\int\limits_{0}^{r}{\frac{{dr}}{{\sqrt{\tilde{f}\left(r\right)\tilde{g}\left(r\right)}}}}}\right)^{-1}\,.$
(3.10)

Plugging the corresponding functions from the MKTN metric (2.10) into the last
equation gives us

$T_{H}=\frac{1}{4}\left({{\mathop{\rm
Im}\nolimits}\int\limits_{0}^{r}{\frac{{\sqrt{\left({r^{2}+a^{2}+l^{2}}\right)^{2}-a^{2}\Delta_{r}}}}{{\left({r-r_{+}}\right)\left({r-r_{-}}\right)}}}}\right)^{-1}\,,$
(3.11)

which yields the Hawking temperature for a MKTN black hole can be written as

$T_{H}=\frac{{r_{+}-r_{-}}}{{4\pi\left({r_{+}^{2}+a^{2}+l^{2}}\right)}}\,.$
(3.12)

As we have expected, this Hawking temperature is the same to that of KTN case
[8]. As an alternative, the Hawking temperature above can also be achieved by
using the straightforward calculation of surface gravity

$\kappa^{2}=-\frac{1}{2}\left({\nabla_{\mu}\xi_{\nu}}\right)\left({\nabla^{\mu}\xi^{\nu}}\right)\,.$
(3.13)

The result is

$\kappa=\frac{{r_{+}-r_{-}}}{{2\left({r_{+}^{2}+a^{2}+l^{2}}\right)}}\,,$
(3.14)

and after using the relation $T_{H}=\tfrac{\kappa}{2\pi}$, we can confirm the
Hawking temperature (3.12) above.

##  4 Microscopic entropy for extremal black holes

In this section, we investigate the Kerr/CFT correspondence for a MKTN black
hole. We would like to extent the previous works [20, 21] where the authors
show how to compute the entropy of an extremal magnetized Kerr black hole
using the Kerr/CFT holography method. The initial step is to find the near
horizon geometry of the extremal black hole which possesses the
$SL\left(2,{\mathbb{R}}\right)\times U\left(1\right)$ isometry. This near
horizon metric together with the corresponding near horizon vector field
solution solve the Eintein-Maxwell equations. Luckily, the general formulation
of asymptotic symmetry group (ASG) method in Kerr/CFT correspondence for the
Einstein-Maxwell theory has been discussed in [33], and had been extended to a
class of gravitational theory in [34]. Hence, in this section we just need to
obtain a near horizon geometry which falls into the category considered in
[33], and employ the general formula for central charge given there.

The near horizon geometry of a MKTN black hole can be achieved by performing
the following coordinate transformation

$t\to\frac{{r_{0}t}}{\lambda}~{}~{},~{}~{}r\to r_{e}+\lambda
r_{0}r~{}~{},~{}~{}\phi\to\phi+\Omega_{J}^{ext}\frac{{r_{0}}}{\lambda}t\,.$
(4.1)

In the equations above we have $\Omega_{J}=\omega^{\prime}$, and
$\Omega_{J}^{ext}$ is the corresponding quantity evaluated at extremality,
i.e. $m^{2}+l^{2}=a^{2}$. By applying this near horizon transformation to the
metric (2.10), one can obtain the near horizon line element

${\rm{d}}s^{2}=\Gamma\left(x\right)\left\\{-r^{2}{\rm{d}}t^{2}+\frac{{{\rm{d}}r^{2}}}{{r^{2}}}+\alpha\left(x\right)\frac{{{\rm{d}}x^{2}}}{{\Delta_{x}}}\right\\}+\gamma\left(x\right)\left({{\rm{d}}\phi+kr{\rm{d}}t}\right)^{2}\,,$
(4.2)

after setting $r_{0}=\sqrt{2}a$. In equation ( 4.2) we have
$\alpha\left(x\right)=1$,

$\Gamma\left(x\right)=a^{-3}\left\\{\left[4\left(4{a}^{8}-{l}^{8}-3{l}^{4}{a}^{4}-5{l}^{6}{a}^{2}+9{l}^{2}{a}^{6}\right)b^{4}-{a}^{4}\left(8{a}^{2}{b}^{2}-1\right)\right]ax^{2}+\left[4\left({l}^{2}{a}^{6}-{l}^{8}+8{a}^{8}\right.\right.\right.$
$\left.\left.\left.-7{l}^{4}{a}^{4}-5{l}^{6}{a}^{2}\right)b^{4}+a^{4}\right]2lx+4a\left(4{a}^{8}-{l}^{8}-3{l}^{4}{a}^{4}-5{l}^{6}{a}^{2}+9{l}^{2}{a}^{6}\right)b^{4}+{a}^{5}\left(8{a}^{2}{b}^{2}+1\right)\right\\}\,,$
(4.3) $\gamma\left(x\right)=\frac{4a^{4}\Delta_{x}}{\Gamma\left(x\right)}\,,$
(4.4)

and

$k=-\frac{\sqrt{a^{2}-l^{2}}}{a^{5}}\left\\{4{a}^{6}{b}^{4}\left(4{a}^{2}+3{l}^{2}\right)-{a}^{4}\left(1-28{b}^{4}{l}^{4}\right)+4{l}^{6}{b}^{4}\left(5{a}^{2}+{l}^{2}\right)\right\\}\,.$
(4.5)

To complete the metric solution above in solving the Einstein-Maxwell
equations, the accompanying vector field is given by

$A_{\mu}{\rm{d}}x^{\mu}=L\left(x\right)\left({{\rm{d}}\phi+kr{\rm{d}}t}\right)\,,$
(4.6)

where

$L\left(x\right)=\frac{L_{1}x+L_{2}}{{\Gamma\left(x\right)}{\left\\{4{b}^{4}\left(9{a}^{6}{l}^{2}-3{a}^{4}{l}^{4}-5{a}^{2}{l}^{6}+4{a}^{8}-{l}^{8}\right)+{a}^{4}\left(1-8{a}^{2}{b}^{2}\right)\right\\}}}\,,$
(4.7)

where

$L_{1}=a\left(36{a}^{6}{b}^{4}{l}^{2}-12{a}^{4}{b}^{4}{l}^{4}-8{a}^{6}{b}^{2}-20{a}^{2}{l}^{6}{b}^{4}+{a}^{4}-4{l}^{8}{b}^{4}+16{a}^{8}{b}^{4}\right)\,,$
(4.8)

and

$L_{2}=l\left(4{a}^{6}{b}^{4}{l}^{2}+32{a}^{8}{b}^{4}-20{a}^{2}{l}^{6}{b}^{4}+{a}^{4}-28{a}^{4}{b}^{4}{l}^{4}-4{l}^{8}{b}^{4}\right)\,.$
(4.9)

As we aimed, this near horizon geometry possesses the
$SL\left(2,{\mathbb{R}}\right)\times U\left(1\right)$ isometry which is vital
for Kerr/CFT correspondence to apply. The $SL\left(2,{\mathbb{R}}\right)$
symmetry is generated by the Killing vectors

$K_{-}=\partial_{t}\,,$ (4.10) $K_{0}=t\partial_{t}-r\partial_{r}\,,$ (4.11)
$K_{+}=\left({\frac{1}{{2r^{2}}}+\frac{{t^{2}}}{2}}\right)\partial_{t}-tr\partial_{r}-\frac{k}{r}\partial_{\phi}\,,$
(4.12)

which obey the commutation relations $\left[{K_{0},K_{\pm}}\right]=\pm
K_{\pm}$ and $\left[{K_{-},K_{+}}\right]=K_{0}$, while the $U\left(1\right)$
symmetry is generated by $\partial_{\phi}$. Here we learn that the presence of
NUT parameter does not break the $SL\left(2,{\mathbb{R}}\right)\times
U\left(1\right)$ symmetry of the near horizon extremal magnetized Kerr black
hole studied in [20, 21].

Moreover, the form of the near horizon metric (4.2) and vector fields (4.6)
match the general forms of near horizon Einstein-Maxwell fields discussed in
[33, 48]. Hence, we can apply the ASG method constructed in [33, 34] to our
present case. Note that we also need to impose the following boundary
condition for the spacetime metric

$h_{\mu\nu}\sim\left({\begin{array}[]{*{20}c}{{\cal
O}\left({r^{2}}\right)}&{{\cal O}\left({r^{1}}\right)}&{{\cal
O}\left({r^{-1}}\right)}&{\cal O}\left(r^{-2}\right)\\\ {}\hfil&{{\cal
O}\left(1\right)}&{{\cal O}\left({r^{-1}}\right)}&{{\cal
O}\left({r^{-1}}\right)}\\\ {}\hfil&{}\hfil&{{\cal
O}\left({r^{-1}}\right)}&{{\cal O}\left({r^{-2}}\right)}\\\
{}\hfil&{}\hfil&{}\hfil&{{\cal O}\left(r^{-3}\right)}\\\
\end{array}}\right)\,,$ (4.13)

and for the Maxwell vector field

$a_{\mu}{\rm{d}}x^{\mu}\sim{\cal O}\left(r\right){\rm{d}}t+{\cal
O}\left({r^{-1}}\right){\rm{d}}r+{\cal O}\left(1\right){\rm{d}}x+{\cal
O}\left(r^{-2}\right){\rm{d}}\phi\,,$ (4.14)

so the formalism examined in [33, 34] can apply. Accordingly, the most general
diffeomorphisms preserving the boundary condition for the spacetime
fluctutation can be written as

$K_{\varepsilon}=\varepsilon\left(\phi\right)\partial_{\phi}-r\frac{{d\varepsilon\left(\phi\right)}}{{d\phi}}\partial_{r}+{\rm{subleadingterm}}$
(4.15)

Now, from the near horizon geometry we have in (4.2), the associated central
charge can be computed using the general formula [33]

$c=c_{grav}+c_{gauge}\,,$ (4.16)

where

$c_{grav}=3k\int\limits_{-1}^{+1}{dx\sqrt{\Gamma\left(x\right)\alpha\left(x\right)\gamma\left(x\right)}}\,,$
(4.17)

and

$c_{gauge}=0.$ (4.18)

Inserting the metric component (4.2) into eq. (4.17) gives us the result

$c=\frac{12m^{2}\left\\{{a}^{4}-4{b}^{4}\left(5{l}^{6}{a}^{2}+7{l}^{4}{a}^{4}+3{l}^{2}{a}^{6}+{l}^{8}+4{a}^{8}\right)\right\\}}{a^{3}}\,.$
(4.19)

As we expected, setting $b=B/2$ and $l=0$ to the last equation gives us the
central charge used in the magnetized Kerr/CFT correspondence investigated in
[20, 21].

Recall that in section 3 we have obtained the Hawking tempereture for a MKTN
black hole, which happens to be the same to the temperature of a KTN black
hole. In extremal state, this Hawking temperature vanishes, as occurred for
the another extremal black holes. However, one can still find the non-
vanishing temperature near the horizon, namely the Frolov-Thorne temperature
obtained by using the formula [34]

$T_{\phi}=\mathop{\lim}\limits_{r_{+}\to
m}\frac{{T_{H}}}{{\Omega_{J}^{{\rm{ext}}}-\Omega_{J}}}=-\left.{\frac{{{{\partial
T_{H}}\mathord{\left/{\vphantom{{\partial T_{H}}{\partial
r_{+}}}}\right.\kern-1.2pt}{\partial
r_{+}}}}}{{{{\partial\Omega_{J}}\mathord{\left/{\vphantom{{\partial\Omega_{J}}{\partial
r_{+}}}}\right.\kern-1.2pt}{\partial r_{+}}}}}}\right|_{r_{+}=m}\,.$ (4.20)

It turns out the Frolov-Thorne temperature for our MKTN case is

$T_{\phi}=\frac{a^{5}}{2\pi\sqrt{a^{2}-l^{2}}\left\\{{a}^{4}-4{b}^{4}\left(5{l}^{6}{a}^{2}+7{l}^{4}{a}^{4}+3{l}^{2}{a}^{6}+{l}^{8}+4{a}^{8}\right)\right\\}}=\frac{1}{2\pi
k}\,,$ (4.21)

where the constant $k$ is given in (4.5).

Now, having obtained the central charged associated to the symmetry of near
horizon of extremal MKTN black hole and the corresponding Frolov-Thorne
temperature, we can proceed to get the entropy using the Cardy formula which
reads

$S_{\rm Cardy}=\frac{\pi^{2}}{3}cT\,.$ (4.22)

After plugging the central charge $c$ in (4.19) and temperature (4.21) into
(4.22), we recover the entropy of extremal MKTN black hole

$S_{\rm ext.}=\frac{{\cal A}_{\rm ext.}}{4}=2\pi a^{2}\,.$ (4.23)

The last equation is just the extremal limit of the entropy given in (3.2),
i.e. after setting $r_{+}=m$ and $m^{2}+l^{2}=a^{2}$. Therefore, here we have
managed to reproduce the entropy of an extremal MKTN black hole by using
microscopic formula (4.22) according to the Kerr/CFT correspondence [32].

##  5 Conclusion and discussion

In this paper we have presented the magnetized version of the KTN spacetime
solution of the Einstein-Maxwell theory. The solution is obtained by employing
the Ernst magnetization introduced in [14], which has been used to magnetized
some known solutions of the Einstein-Maxwell equations. The novel solution
presented in this paper can be considered as a Taub-NUT extension of the
previous solution known in literature as the magnetized Kerr spacetime [14].
Some aspects of the spacetime are discussed, namely the area of black hole,
squared of Riemann tensor on equator, and the corresponding Hawking
temperature.

In addition to the studies on some aspects of the new solution, we also
discussed the Kerr/CFT holography associated to the spacetime. The agreement
of near horizon metric (4.2) to the general form discussed in [33, 34] allows
us to employ the method developed in that works to obtain the corresponding
central charge and Frolov-Thorne temperature. In turn, these quantities give
us the extremal black hole entropy after using the Cardy formula, which can be
viewed as the microscopic calculation for the black hole entropy. This result
is a generalization of some previous works on the magnetized Kerr/CFT
correspondence [20, 21], where now the spacetime is equipped with the NUT
parameter.

There are several future projects that can be pursued based on the results
presented in this paper. First is to extend the solution obtained here to the
charged one, namely the magnetized Kerr-Newman-Taub-NUT solution [50]. The
method would be similar, but the incorporating functions and the results would
be more complicated compared to those appearing in this paper. Also discussing
the associated Meissner effect as that investigated in [19] for an extremal
magnetized black holes with NUT parameter would be interesting.

##  Appendix A Ernst magnetization

In this appendix, let us review briefly the Ernst magnetization procedure
employed in section 2. The seed metric should be expressible in the form of
Lewis-Papapetrou-Weyl (LPW) line element, namely

$ds^{2}=f\left({d\phi-\omega
dt}\right)^{2}-f^{-1}\left({\rho^{2}dt^{2}-e^{2\gamma}d\zeta
d\zeta^{*}}\right)\,.$ (A.1)

Above, $f$, $\gamma$, and $\omega$ are functions of $\zeta$. Moreover, we have
used $-+++$ sign convention for the spacetime, and ∗ notation represents the
complex conjugation. Together with the vector solution
$A_{\mu}dx^{\mu}=A_{t}dt+A_{\phi}d\phi$, the line element above construct the
Ernst potentials

$\Phi=A_{\phi}+i\tilde{A}_{\phi}\,,$ (A.2)

and

${\cal E}=f+\left|{\Phi}\right|^{2}-i\Psi\,.$ (A.3)

The first potential above is known as the electromagnetic one, and the latter
one is gravitational potential. Particularly for the electromagnetic potential
(A.2), the imaginary part is obtained by solving

$\nabla A_{t}+\omega\nabla A_{\phi}=-i\frac{\rho}{f}\nabla\tilde{A}_{\phi}\,.$
(A.4)

On the other hand, the twist potential $\Psi$ in (A.3) obeys the differential
equations

$\nabla\Psi=\frac{{if^{2}}}{\rho}\nabla\omega+2i\Phi^{*}\nabla\Phi\,.$ (A.5)

In terms of Ernst potentials (A.3) and (A.2), the Einstein-Maxwell field
equations can be written as

$\left({{\mathop{\rm Re}\nolimits}\left\\{{\cal
E}\right\\}+{{\left|\Phi\right|}^{2}}}\right)\nabla{\cal E}=\left({\nabla{\cal
E}+2{\Phi^{*}}\nabla\Phi}\right)\cdot\nabla{\cal E}\,,$ (A.6)

and

$\left({{\mathop{\rm Re}\nolimits}\left\\{{\cal
E}\right\\}+{{\left|\Phi\right|}^{2}}}\right)\nabla{\Phi}=\left({\nabla{\cal
E}+2{\Phi^{*}}\nabla\Phi}\right)\cdot\nabla{\Phi}\,,$ (A.7)

which are known as the Ernst equations. The last two equations are invariant
under a type of Harrison transformation

${\cal E}\to{\cal E}^{\prime}=\Lambda^{-1}{\cal E}~{}~{}~{}{\rm
and}~{}~{}~{}\Phi\to\Phi^{\prime}=\Lambda^{-1}\left({\Phi-b{\cal
E}}\right)\,,$ (A.8)

where

$\Lambda=1-2b\Phi+b^{2}{\cal E}\,.$ (A.9)

The constant $b$ in the equation above represent the strength of external
magnetic field where the black hole is embedded.

Furthermore, the magnetized line element (A.1) resulting from the
magnetization (A.8) will contain

$f^{\prime}={\mathop{\rm Re}\nolimits}\left\\{{{\cal
E}^{\prime}}\right\\}-\left|{\Phi^{\prime}}\right|^{2}=\left|\Lambda\right|^{-2}f\,,$
(A.10)

and

$\nabla\omega^{\prime}=\left|\Lambda\right|^{2}\nabla\omega-\frac{\rho}{f}\left({\Lambda^{*}\nabla\Lambda-\Lambda\nabla\Lambda^{*}}\right)\,,$
(A.11)

while the function $\gamma$ remains unaffected. A little bit more detail,
since we would like to express the metric in a Boyer-Lindquist type, the term
$d\zeta d\zeta^{*}$ in (A.1) can be written as

$d\zeta
d\zeta^{*}=\frac{{dr^{2}}}{{\Delta_{r}}}+\frac{{dx^{2}}}{{\Delta_{x}}}\,,$
(A.12)

where $\Delta_{r}=\Delta_{r}\left(r\right)$ and
$\Delta_{x}=\Delta_{x}\left(x\right)$. Note that the corresponding operator
$\nabla$ would take the form
$\nabla=\sqrt{\Delta_{r}}\partial_{r}+i\sqrt{\Delta_{x}}\partial_{x}$, and we
would have $\rho^{2}=\Delta_{r}\Delta_{x}$. Consequently, (A.4) gives us

$\partial_{r}A_{t}=-\omega\partial_{r}A_{\phi}+\frac{{\Delta_{x}}}{f}\partial_{x}\tilde{A}_{\phi}\,,$
(A.13)

and

$\partial_{x}A_{t}=-\omega\partial_{x}A_{\phi}-\frac{{\Delta_{r}}}{f}\partial_{r}\tilde{A}_{\phi}\,.$
(A.14)

The last two equations are useful to obtain the $A_{t}$ component associated
to the magnetized spacetime according to (A.8). To end this appendix, another
equations which are important to complete the metric solution are

$\partial_{r}\omega^{\prime}=\left|\Lambda\right|^{2}\partial_{r}\omega+\frac{{\Delta_{x}}}{f}{\mathop{\rm
Im}\nolimits}\left\\{{\Lambda^{*}\partial_{x}\Lambda-\Lambda\partial_{x}\Lambda^{*}}\right\\}\,,$
(A.15)

and

$\partial_{x}\omega^{\prime}=\left|\Lambda\right|^{2}\partial_{x}\omega-\frac{{\Delta_{r}}}{f}{\mathop{\rm
Im}\nolimits}\left\\{{\Lambda^{*}\partial_{r}\Lambda-\Lambda\partial_{r}\Lambda^{*}}\right\\}\,.$
(A.16)

##  Appendix B Components of electromagnetic potential

Here we present the component of electromagnetic Ernst potential
$\Phi^{\prime}=A_{\phi}+i{\tilde{A}}_{\phi}$ in (2.12). The real part reads

${A_{\phi}}={b~{}{\sum\limits_{j=0}^{6}{{a_{j}}{x^{j}}}}\over{\sum\limits_{k=0}^{4}{{b_{k}}{x^{k}}}}}$
(B.1)

where the function $a_{j}$’s are

$a_{6}=-a^{6}b^{2}\Delta_{r}^{2}\,,$ $a_{5}=6a^{5}b^{3}l\Delta_{r}^{2}\,,$
$a_{4}=a^{4}\left\\{-{r}^{6}{b}^{2}-15{b}^{2}{r}^{4}{l}^{2}-4{b}^{2}m\left(3{a}^{2}-10{l}^{2}\right)r^{3}+\left(24{b}^{2}{a}^{2}{m}^{2}-36{b}^{2}{l}^{2}{m}^{2}+9{b}^{2}{l}^{4}\right.\right.$
$\left.-36{b}^{2}{a}^{2}{l}^{2}+3{b}^{2}{a}^{4}+{a}^{2}\right)r^{2}+-2m\left(12{b}^{2}{l}^{4}+{a}^{2}-42{b}^{2}{a}^{2}{l}^{2}+6{b}^{2}{a}^{4}\right)r$
$\left.-4\,{b}^{2}\left({l}^{2}+{a}^{2}\right)^{2}{m}^{2}+28{b}^{2}{a}^{2}{l}^{4}-9{b}^{2}{l}^{6}+2{a}^{6}{b}^{2}+{a}^{4}-21{b}^{2}{a}^{4}{l}^{2}-{l}^{2}{a}^{2}\right\\}\,,$
$a_{3}=4a^{3}l\left\\{3{b}^{2}{a}^{2}{r}^{4}-12{b}^{2}{a}^{2}m{r}^{3}\left(6{b}^{2}{a}^{4}-2{b}^{2}{a}^{2}{l}^{2}+22{b}^{2}{a}^{2}{m}^{2}+{a}^{2}+2{b}^{2}{l}^{2}{m}^{2}\right){r}^{2}\right.$
$\left.-2m\left(10{b}^{2}{a}^{4}-12{b}^{2}{a}^{2}{l}^{2}+{a}^{2}-2{b}^{2}{l}^{4}\right)r-2\,{b}^{2}\left({l}^{2}+{a}^{2}\right)^{2}{m}^{2}-{l}^{2}{a}^{2}+{a}^{4}-10{b}^{2}{a}^{4}{l}^{2}+7{b}^{2}{a}^{2}{l}^{4}+3{a}^{6}{b}^{2}\right\\}\,,$
$a_{2}=a^{2}\left\\{2a^{2}b^{2}r^{6}+{a}^{2}\left(6{l}^{2}{b}^{2}+1+3{b}^{2}{a}^{2}\right)r^{4}-4{b}^{2}m\left(2{l}^{4}-3{a}^{4}+2{l}^{2}{a}^{2}\right)r^{3}+\left(6{l}^{2}{a}^{2}+24{b}^{2}{m}^{2}{l}^{4}\right.\right.$
$\left.+18{b}^{2}{a}^{4}{l}^{2}+48{a}^{2}{b}^{2}{l}^{2}{m}^{2}-36{b}^{2}{a}^{4}{m}^{2}+30{b}^{2}{a}^{2}{l}^{4}\right)r^{2}+4m\left(3{a}^{6}{b}^{2}+6{b}^{2}{l}^{6}+{a}^{4}-33{b}^{2}{a}^{4}{l}^{2}-2{l}^{2}{a}^{2}\right)r$
$\left.-4{b}^{2}\left(2{a}^{2}+3{l}^{2}\right)\left({l}^{2}+{a}^{2}\right)^{2}m^{2}-{a}^{2}\left(37{b}^{2}{a}^{2}{l}^{4}-28{b}^{2}{a}^{4}{l}^{2}+{a}^{6}{b}^{2}+3{l}^{4}+{a}^{4}-8{l}^{2}{a}^{2}+6{b}^{2}{l}^{6}\right)\right\\}\,,$
$a_{1}=-2la\left\\{3{a}^{4}{b}^{2}{r}^{4}+4{a}^{4}{r}^{3}{b}^{2}m+2{a}^{2}\left(6{b}^{2}{l}^{2}{m}^{2}+12{b}^{2}{a}^{2}{m}^{2}+3{b}^{2}{a}^{4}+{a}^{2}+9{b}^{2}{a}^{2}{l}^{2}\right)r^{2}\right.$
$\left.-4m{a}^{2}\left({a}^{2}-9{b}^{2}{a}^{2}{l}^{2}-6{b}^{2}{l}^{4}+3{b}^{2}{a}^{4}\right)r+4{b}^{2}\left({l}^{2}+{a}^{2}\right)^{3}{m}^{2}+{a}^{4}\left(2{a}^{2}-5{b}^{2}{l}^{4}-2{l}^{2}-14{b}^{2}{a}^{2}{l}^{2}+3{b}^{2}{a}^{4}\right)\right\\}\,,$
$a_{0}=-\left\\{a^{4}b^{2}r^{6}+{a}^{4}\left(1+7{l}^{2}{b}^{2}+2{b}^{2}{a}^{2}\right)r^{4}+4\,{b}^{2}{a}^{2}m\left({a}^{4}+4\,{l}^{2}{a}^{2}+2\,{l}^{4}\right)r^{3}+\left(16{a}^{4}{b}^{2}{m}^{2}{l}^{2}+{a}^{8}{b}^{2}\right.\right.$
$\left.+16{a}^{2}{b}^{2}{m}^{2}{l}^{4}+7{a}^{4}{b}^{2}{l}^{4}+{a}^{6}+2{l}^{2}{a}^{4}+4{m}^{2}{l}^{6}{b}^{2}\right)r^{2}+2m{a}^{2}\left(2{a}^{6}{b}^{2}+4{b}^{2}{a}^{2}{l}^{4}+4{b}^{2}{l}^{6}-2{b}^{2}{a}^{4}{l}^{2}+{a}^{4}\right)r$
$\left.4{b}^{2}\left({l}^{2}+{a}^{2}\right)^{4}{m}^{2}+{l}^{2}{a}^{4}\left(3{a}^{2}+{l}^{2}\right)\left(3{b}^{2}{a}^{2}+{l}^{2}{b}^{2}+1\right)\right\\}\,,$

and the corresponding $b_{k}$’s are

$b_{6}=b^{4}a^{6}\Delta_{r}^{2}\,,$ $b_{5}=6la^{5}b^{4}\Delta_{r}^{2}\,,$
$b_{4}=-b^{2}a^{4}\left\\{2{a}^{6}{b}^{2}+2{a}^{4}+84r{b}^{2}m{l}^{2}{a}^{2}+40{b}^{2}m{r}^{3}{l}^{2}-24{b}^{2}mr{l}^{4}-36{m}^{2}{l}^{2}{r}^{2}{b}^{2}-15{r}^{4}{b}^{2}{l}^{2}\right.$
$+9{l}^{4}{b}^{2}{r}^{2}-4{b}^{2}{m}^{2}{l}^{4}-12{b}^{2}{a}^{4}mr-4mr{a}^{2}-36{b}^{2}{a}^{2}{r}^{2}{l}^{2}+24{b}^{2}{a}^{2}{m}^{2}{r}^{2}-8{a}^{2}{b}^{2}{l}^{2}{m}^{2}-12{b}^{2}{a}^{2}m{r}^{3}$
$\left.-4{b}^{2}{a}^{4}{m}^{2}-21{b}^{2}{a}^{4}{l}^{2}+3{b}^{2}{a}^{4}{r}^{2}+2{r}^{2}{a}^{2}-2{l}^{2}{a}^{2}+28{b}^{2}{a}^{2}{l}^{4}-9{l}^{6}{b}^{2}-{b}^{2}{r}^{6}\right\\}\,,$
$b_{3}=4b^{2}la^{3}\left\\{20{b}^{2}{a}^{4}mr+4mr{a}^{2}+12{b}^{2}{a}^{2}m{r}^{3}-24r{b}^{2}m{l}^{2}{a}^{2}-22{b}^{2}{a}^{2}{m}^{2}{r}^{2}-4{b}^{2}mr{l}^{4}\right.$
$-2{m}^{2}{l}^{2}{r}^{2}{b}^{2}-3{b}^{2}{a}^{2}{r}^{4}+2{b}^{2}{a}^{2}{r}^{2}{l}^{2}+4{a}^{2}{b}^{2}{l}^{2}{m}^{2}+2{b}^{2}{m}^{2}{l}^{4}-6{b}^{2}{a}^{4}{r}^{2}+2{b}^{2}{a}^{4}{m}^{2}$
$\left.-2{r}^{2}{a}^{2}-3{a}^{6}{b}^{2}-2{a}^{4}+10{b}^{2}{a}^{4}{l}^{2}+2{l}^{2}{a}^{2}-7{b}^{2}{a}^{2}{l}^{4}\right\\}\,,$
$b_{2}=a^{2}\left\\{6{b}^{2}{a}^{2}{l}^{4}-2{b}^{2}{a}^{2}{r}^{4}-16{b}^{2}{a}^{4}{l}^{2}-28{a}^{6}{b}^{4}{l}^{2}+{a}^{8}{b}^{4}-12r{a}^{6}{b}^{4}m+132r{l}^{2}{b}^{4}{a}^{4}m+16r{b}^{2}m{l}^{2}{a}^{2}\right.$
$-24{b}^{4}{m}^{2}{l}^{4}{r}^{2}-6{b}^{4}{r}^{4}{l}^{2}{a}^{2}+36{b}^{4}{a}^{4}{m}^{2}{r}^{2}-2{b}^{4}{r}^{6}{a}^{2}-3{r}^{4}{a}^{4}{b}^{4}+8{b}^{4}{r}^{3}{l}^{2}{a}^{2}m+2{a}^{6}{b}^{2}-48{a}^{2}{b}^{4}{m}^{2}{l}^{2}{r}^{2}$
$-12{b}^{2}{a}^{2}{r}^{2}{l}^{2}-8{b}^{2}{a}^{4}mr-12{r}^{3}{b}^{4}{a}^{4}m+{a}^{4}+12{b}^{4}{m}^{2}{l}^{6}+32{b}^{4}{m}^{2}{l}^{4}{a}^{2}+28{b}^{4}{m}^{2}{a}^{4}{l}^{2}$
$\left.+8{a}^{6}{b}^{4}{m}^{2}-24{l}^{6}{b}^{4}mr+8{l}^{4}{b}^{4}m{r}^{3}-30{b}^{4}{a}^{2}{r}^{2}{l}^{4}-18{b}^{4}{a}^{4}{r}^{2}{l}^{2}+6{b}^{4}{l}^{6}{a}^{2}+37{b}^{4}{a}^{4}{l}^{4}\right\\}\,,$
$b_{1}=2la\left\\{{a}^{4}+4{b}^{4}{m}^{2}{l}^{6}-12r{a}^{6}{b}^{4}m-8{b}^{2}{a}^{4}mr+24{b}^{4}{a}^{4}{m}^{2}{r}^{2}+36r{l}^{2}{b}^{4}{a}^{4}m+4{r}^{3}{b}^{4}{a}^{4}m+24{b}^{4}{l}^{4}{a}^{2}mr\right.$
$+12{a}^{2}{b}^{4}{m}^{2}{l}^{2}{r}^{2}+3{r}^{4}{a}^{4}{b}^{4}+18{b}^{4}{a}^{4}{r}^{2}{l}^{2}+12{b}^{4}{m}^{2}{a}^{4}{l}^{2}+12{b}^{4}{m}^{2}{l}^{4}{a}^{2}+6{r}^{2}{b}^{4}{a}^{6}+4{a}^{6}{b}^{4}{m}^{2}$
$\left.+4{b}^{2}{a}^{4}{r}^{2}+3{a}^{8}{b}^{4}+4{a}^{6}{b}^{2}-14{a}^{6}{b}^{4}{l}^{2}-4{b}^{2}{a}^{4}{l}^{2}-5{b}^{4}{a}^{4}{l}^{4}\right\\}\,,$
$b_{0}=4{b}^{4}{m}^{2}{l}^{8}+{b}^{4}\left(8mr{a}^{2}+{a}^{4}+4{m}^{2}{r}^{2}+16{a}^{2}{m}^{2}\right){l}^{6}+{b}^{2}{a}^{2}\left(7{r}^{2}{a}^{2}{b}^{2}+8{b}^{2}m{r}^{3}+2{a}^{2}\right.$
$\left.+8{b}^{2}{a}^{2}mr+6{b}^{2}{a}^{4}+16{b}^{2}{m}^{2}{r}^{2}+24{b}^{2}{a}^{2}{m}^{2}\right){l}^{4}+{a}^{4}\left(16{b}^{4}m{r}^{3}+9{b}^{4}{a}^{4}+6{b}^{2}{a}^{2}+4{b}^{2}{r}^{2}+7{b}^{4}{r}^{4}\right.$
$\left.+16{b}^{4}{m}^{2}{r}^{2}-4r{b}^{4}{a}^{2}m+16{b}^{4}{m}^{2}{a}^{2}+1\right){l}^{2}+{a}^{4}\left(r+{b}^{2}{a}^{2}r+{b}^{2}{r}^{3}+2{b}^{2}{a}^{2}m\right)^{2}\,.$

The imaginary one can be written as

${{\tilde{A}}_{\phi}}={{-2a^{2}b}~{}{\sum\limits_{j=0}^{3}{{{\tilde{a}}_{j}}{x^{j}}}}\over{\sum\limits_{k=0}^{6}{{{\tilde{b}}_{k}}{x^{k}}}}}\,,$
(B.2)

where we have ${\tilde{a}}_{j}$’s as the followings

${\tilde{a}}_{3}={a}^{3}\left(m{a}^{2}-m{r}^{2}+m{l}^{2}-2r{l}^{2}\right)\,,$
${\tilde{a}}_{2}={a}^{2}l\left({r}^{3}-3m{r}^{2}+2m{l}^{2}-3r{l}^{2}+2m{a}^{2}+3r{a}^{2}\right)\,,$
${\tilde{a}}_{1}=a\left(3\,{a}^{2}m{r}^{2}+{a}^{4}m+3\,{a}^{2}m{l}^{2}+2m{l}^{4}+6{a}^{2}r{l}^{2}\right)\,,$
${\tilde{a}}_{0}=l\left({a}^{2}r{l}^{2}+{a}^{2}{r}^{3}+{a}^{4}m+2{a}^{2}m{l}^{2}+m{l}^{4}+2{a}^{2}m{r}^{2}+{l}^{2}m{r}^{2}-{a}^{4}r\right)\,,$

and the associated ${\tilde{b}}_{k}$’s read

${\tilde{b}}_{6}=b^{4}a^{6}\Delta_{r}^{2}\,,$
${\tilde{b}}_{5}=6la^{5}b^{4}\Delta_{r}^{2}\,,$
${\tilde{b}}_{4}=-b^{2}a^{4}\left\\{2{a}^{6}{b}^{2}+2{a}^{4}+84r{b}^{2}m{l}^{2}{a}^{2}+40{b}^{2}m{r}^{3}{l}^{2}-24{b}^{2}mr{l}^{4}-36{m}^{2}{l}^{2}{r}^{2}{b}^{2}-15{r}^{4}{b}^{2}{l}^{2}\right.$
$+9{l}^{4}{b}^{2}{r}^{2}-4{b}^{2}{m}^{2}{l}^{4}-12{b}^{2}{a}^{4}mr-4mr{a}^{2}-36{b}^{2}{a}^{2}{r}^{2}{l}^{2}+24{b}^{2}{a}^{2}{m}^{2}{r}^{2}-8{a}^{2}{b}^{2}{l}^{2}{m}^{2}-12{b}^{2}{a}^{2}m{r}^{3}$
$\left.-4{b}^{2}{a}^{4}{m}^{2}-21{b}^{2}{a}^{4}{l}^{2}+3{b}^{2}{a}^{4}{r}^{2}+2{r}^{2}{a}^{2}-2{l}^{2}{a}^{2}+28{b}^{2}{a}^{2}{l}^{4}-9{l}^{6}{b}^{2}-{b}^{2}{r}^{6}\right\\}\,,$
${\tilde{b}}_{3}=-4b^{2}la^{3}\left\\{-20{b}^{2}{a}^{4}mr-4mr{a}^{2}-12{b}^{2}{a}^{2}m{r}^{3}+24r{b}^{2}m{l}^{2}{a}^{2}+22{b}^{2}{a}^{2}{m}^{2}{r}^{2}+4{b}^{2}mr{l}^{4}\right.$
$+2{m}^{2}{l}^{2}{r}^{2}{b}^{2}+3{b}^{2}{a}^{2}{r}^{4}-2{b}^{2}{a}^{2}{r}^{2}{l}^{2}-4{a}^{2}{b}^{2}{l}^{2}{m}^{2}-2{b}^{2}{m}^{2}{l}^{4}+6{b}^{2}{a}^{4}{r}^{2}-2{b}^{2}{a}^{4}{m}^{2}$
$\left.+2{r}^{2}{a}^{2}+3{a}^{6}{b}^{2}+2{a}^{4}-10{b}^{2}{a}^{4}{l}^{2}-2{l}^{2}{a}^{2}+7{b}^{2}{a}^{2}{l}^{4}\right\\}\,,$
${\tilde{b}}_{2}=a^{2}\left\\{6{b}^{2}{a}^{2}{l}^{4}-2{b}^{2}{a}^{2}{r}^{4}-16{b}^{2}{a}^{4}{l}^{2}-28{a}^{6}{b}^{4}{l}^{2}+{a}^{8}{b}^{4}-12r{a}^{6}{b}^{4}m+132r{l}^{2}{b}^{4}{a}^{4}m+16r{b}^{2}m{l}^{2}{a}^{2}\right.$
$-24{b}^{4}{m}^{2}{l}^{4}{r}^{2}-6{b}^{4}{r}^{4}{l}^{2}{a}^{2}+36{b}^{4}{a}^{4}{m}^{2}{r}^{2}-2{b}^{4}{r}^{6}{a}^{2}-3{r}^{4}{a}^{4}{b}^{4}+8{b}^{4}{r}^{3}{l}^{2}{a}^{2}m+2{a}^{6}{b}^{2}-48{a}^{2}{b}^{4}{m}^{2}{l}^{2}{r}^{2}$
$-12{b}^{2}{a}^{2}{r}^{2}{l}^{2}-8{b}^{2}{a}^{4}mr-12{r}^{3}{b}^{4}{a}^{4}m+{a}^{4}+12{b}^{4}{m}^{2}{l}^{6}+32{b}^{4}{m}^{2}{l}^{4}{a}^{2}+28{b}^{4}{m}^{2}{a}^{4}{l}^{2}+8{a}^{6}{b}^{4}{m}^{2}$
$\left.-24{l}^{6}{b}^{4}mr+8{l}^{4}{b}^{4}m{r}^{3}-30{b}^{4}{a}^{2}{r}^{2}{l}^{4}-18{b}^{4}{a}^{4}{r}^{2}{l}^{2}+6{b}^{4}{l}^{6}{a}^{2}+37{b}^{4}{a}^{4}{l}^{4}\right\\}\,,$
${\tilde{b}}_{1}=2la\left\\{{a}^{4}+4{b}^{4}{m}^{2}{l}^{6}-12r{a}^{6}{b}^{4}m-8{b}^{2}{a}^{4}mr+24{b}^{4}{a}^{4}{m}^{2}{r}^{2}+36r{l}^{2}{b}^{4}{a}^{4}m+4{r}^{3}{b}^{4}{a}^{4}m\right.$
$+24{b}^{4}{l}^{4}{a}^{2}mr+12{a}^{2}{b}^{4}{m}^{2}{l}^{2}{r}^{2}+3{r}^{4}{a}^{4}{b}^{4}+18{b}^{4}{a}^{4}{r}^{2}{l}^{2}+12{b}^{4}{m}^{2}{a}^{4}{l}^{2}+12{b}^{4}{m}^{2}{l}^{4}{a}^{2}+6{r}^{2}{b}^{4}{a}^{6}$
$\left.+4{a}^{6}{b}^{4}{m}^{2}+4{b}^{2}{a}^{4}{r}^{2}+3{a}^{8}{b}^{4}+4{a}^{6}{b}^{2}-14{a}^{6}{b}^{4}{l}^{2}-4{b}^{2}{a}^{4}{l}^{2}-5{b}^{4}{a}^{4}{l}^{4}\right\\}\,,$
${\tilde{b}}_{0}=4{b}^{4}{m}^{2}{l}^{8}+{b}^{4}\left(8mr{a}^{2}+{a}^{4}+4{m}^{2}{r}^{2}+16{a}^{2}{m}^{2}\right){l}^{6}+{b}^{2}{a}^{2}\left(7{r}^{2}{a}^{2}{b}^{2}+8{b}^{2}m{r}^{3}+2{a}^{2}\right.$
$\left.+8{b}^{2}{a}^{2}mr+6{b}^{2}{a}^{4}+16{b}^{2}{m}^{2}{r}^{2}+24{b}^{2}{a}^{2}{m}^{2}\right){l}^{4}+{a}^{4}\left(16{b}^{4}m{r}^{3}+9{b}^{4}{a}^{4}+6{b}^{2}{a}^{2}+4{b}^{2}{r}^{2}+7{b}^{4}{r}^{4}\right.$
$\left.+16{b}^{4}{m}^{2}{r}^{2}-4r{b}^{4}{a}^{2}m+16{b}^{4}{m}^{2}{a}^{2}+1\right){l}^{2}+{a}^{4}\left(r+{b}^{2}{a}^{2}r+{b}^{2}{r}^{3}+2{b}^{2}{a}^{2}m\right)^{2}\,.$

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11institutetext:  Department of Physical Sciences, Indian Institute of Science
Education and Research Kolkata, West Bengal 741246, India
Center for Quantum Science and Technology, Siksha ’O’ Anusandhan University,
Bhubaneswar, Odisha 751030, India
Department of Physics, Savitribai Phule Pune University, Pune 411007, India

# Neutral-atom qubits in atom-molecular BEC

Leena Barshilia 11 Rajiuddin Sk 1122 Prasanta K. Panigrahi 1122 Avinash Khare
33112233

###### Abstract

Recently, neutral atoms have emerged as a promising platform for quantum
computing, offering scalability. In this study, we showcase the realization of
atomic qubits in atom-molecular Bose-Einstein condensate, belonging to three
distinct classes. In the first case, the condensed molecules form a droplet
platform with a flat-top configuration, facilitating effective isolation from
both external environments and neighbouring molecules. The second atomic
qubits have wavefunctions in the “pulse” form, exhibiting power law behaviour,
whereas the third one has ground and excited state wavefunctions in their
respective composite forms, $\sech^{2}{\beta x}$ and $\sech{\beta
x}\tanh{\beta x}$. The localization of the qubits depends on the chemical
potential, which is governed by the photo association, providing effective
control for qubit manipulation. The relevant parameters, such as energy level
separation, healing length, and atom numbers, are found to be influenced by
the non-linearity and strength of photo associations governing the behaviour
of macroscopic qubits and molecular droplets.

##  1 Introduction

In recent years, significant advancements have been achieved in the
development of quantum computers, utilizing quantum states and operations to
encode and process information. A quantum bit is fragile and very much
susceptible to interactions with the environment. Therefore, a good quality
qubit with a long coherence time and less sensitive to the environment is the
need of the hour. Qubits can be realized through a variety of physical
systems. Examples include trapped ions [1, 2], superconducting setups [3],
quantum dots [4], and optical processors [5]. Apart from these physical
systems, the array of neutral atoms controlled by beams of light has emerged
as a highly potent and scalable technology for manipulating quantum registers
containing up to several thousand qubits [6, 7]. Both for superconducting
circuits or silicon spin qubits, the challenge lies in manufacturing
artificial atoms that are exceedingly difficult to make identical. In
contrast, neutral atoms are inherently identical, a crucial factor for
enabling efficient quantum computation. In this connection, it is of deep
interest to investigate the potential of neutral atoms, which are based on
Bose-Einstein condensate (BEC) and readily realizable experimentally [8].
Atomic BEC has been recently investigated for this possibility. The BEC and
defect complex have been shown to admit qubit and, in general, qudit states
for the defect in the presence of kink solitons for atomic BEC [9]. The
dynamics of the two-component BEC are captured by coupled nonlinear
Schrödinger equations (NLSE), exhibiting paired solitons [8]. The fact that
soliton can realize logical states was proposed by Laxmanan and Radhakrishnan
in optical fibre, governed by an integrable coupled two-component nonlinear
Schrödinger equation [10]. As the solitons of this system can have both
elastic and non-elastic scattering, these extended objects can be fully
controlled, which has been subsequently physically realized. These dynamics of
two-component BEC include a limiting form introduced by Petrov [11],
particularly in the context of Bose condensates with Lee-Huang-Yang quantum
corrections incorporated via a non-analytic quadratic non-linearity. Bose-
Einstein condensates (BECs) can be confined to behave as one-dimensional and
two-dimensional systems through a harmonic trap configuration. The possibility
of realizing qubits, with both grey soliton and stable bright soliton, has
been considered [12, 13]. It is worth mentioning that the presence of both
dark and bright solitons leads to the well-known Pöschl-Teller potential,
which is reflectionless in the appropriate parameter domain and solvable,
thereby yielding exact solutions for the qudit system [14]. In the presence of
grey solitons, all-optical controlled-NOT gate implementation has been shown
through appropriate reflectionless potential. As is well known, the
superconducting qubits underlying most of the present quantum computing
platforms are derived from the sine Gordon model, describing the
superconducting order parameter [15].

The study of atom-molecular Bose-Einstein condensates (AMBEC) has been a focus
of research, especially in the realm of cold chemistry [16, 17]. Cold
molecules, with their complex energy structure, hold the potential to drive
advancements in quantum engineering and quantum chemistry [18, 19, 20]. Due to
the difficulty in controlling the cooling of molecules, there has been
significant interest in exploring molecular condensation via the atomic route,
which begins with an atomic quantum gas and then pairs the atoms into
molecules [21]. It is of deep interest to investigate the possibility of
realizing a quantum computing platform using this atom molecular BEC. These
systems have physically attracted attention due to the macroscopic nature of
the wavefunctions and robustness against decoherence. Recently, quantum
molecular devices have attracted significant attention for quantum information
processing due to their compact size, which helps in maintaining coherence
over many operations [22].

In this paper, we demonstrate that two orthogonal solutions of atomic and
molecular mean field equations can be effectively conceptualized as a ground
state and an excited state, thereby forming a qubit system. We find a wide
class of exact two-level systems for the AMBEC complex belonging to the atomic
level, which may possibly find an application as qubits. These include Pöschl-
Teller type solutions, where the molecule can be a quantum droplet and lump-
type or bright, dark composite solitons. The spatial spread of power law
nature indicates localization in the frequency domain. Furthermore,
“molecular” soliton-type configurations are also found to realize qubits where
the molecular condensate is a bright soliton. Additionally, we outline the
physical operations that can be regarded as quantum gates for quantum
computation tasks.

##  2 Model

In recent years, significant advancements have propelled Photoassociation to
the forefront of research in the realm of ultracold atomic gases [23, 24].
This technique involves the formation of molecules from two atoms through the
absorption of a photon from an applied optical field during atomic collisions.
The development of experimental capabilities has enabled precise quantum
control over the intricate internal degrees of freedom within molecules [25,
26, 27], thereby creating a unique platform for cold and ultracold chemistry.

Originating from early investigations into ultracold atom collisions [17, 28],
recent efforts have focused on engineering more complex reactions in the
quantum regime. Here, phenomena such as quantum threshold laws, quantum
statistics, and single partial wave scattering govern collision processes [29,
30]. These processes are controlled by external electric and magnetic fields.
To describe control mechanism in photoassociation, we focus on diatomic
molecular formation, which represents the most elementary second-order
reaction:

$A+A\Leftrightarrow A_{2}$.

In this scenario, the formation of potential products results in various
outcomes as dictated by quantum statistics: $bb\rightarrow b$, $bf\rightarrow
f$, and $ff\rightarrow b$, where $b$ represents bosonic and $f$ denotes
fermionic counterparts.

Our focus lies on a chemical system of the former type, where bosonic
enhancement exerts the most significant influence on chemical dynamics.

Order | Reaction | $\hat{H}_{\text{int}}$
---|---|---
0. | $\text{Bath}\Leftrightarrow A$ | $k(\hat{a}_{A}^{\dagger}+\text{h.c.})$
1. | $A\Leftrightarrow B$ | $k(\hat{a}^{\dagger}_{A}\hat{a}_{B}+\text{h.c.})$
2. | $A+A\Leftrightarrow A_{2}$ | $k(\hat{a}_{A}^{\dagger}\hat{a}_{A}^{\dagger}\hat{a}_{A_{2}}+\text{h.c.})$
Table 1: The proposed interaction Hamiltonians [31] for low-order bosonic
reactions.

In Table 1, various orders of chemical reactions are depicted within a broad
context. A zeroth-order reaction represents the exchange of species with a
reservoir; on the other hand, a first-order reaction characterizes a linear
interaction between two quantum fields. Conversely, atom-molecule
interconversion necessitates a Hamiltonian incorporating a second-order
reaction.

$\hat{H}=E_{A}\hat{n}_{A}+E_{A_{2}}\hat{n}_{A_{2}}+k(\hat{a}_{A}^{\dagger}\hat{a}_{A}^{\dagger}\hat{a}_{A_{2}}+h.c.),$
(1)

where $E_{A}$ and $E_{A_{2}}$ label the corresponding ground-state
energies.This Hamiltonian can be extended to encompass the influence of
multiple concurrent reactions, particle loss, and dissipation. In our work, we
disregard these effects and assume that the reaction rate significantly
exceeds the ground state energies, i.e., $k\gg|E_{A}|+|E_{A_{2}}|$. It is
important to note that within this framework, we describe reversible reactions
with a single reaction rate. Furthermore, this model provides insights into
the outcome of a chemical reaction without directly revealing the underlying
processes of bond breaking and formation.

The system’s Hamiltonian can be expressed using field operators for both atoms
and the molecular resonant state. When resonance occurs, the quantity of
molecules increases significantly, culminating in the creation of a molecular
Bose-Einstein condensate (BEC). Our considerations encompass two-body
interactions involving atom-atom, molecule-molecule, and atom-molecule
collisions, alongside the component accountable for the conversion of pairs of
atoms into molecules and vice versa, in terms of natural unit.

$\displaystyle\hat{H}$ $\displaystyle=$ $\displaystyle\large{\int}{\rm
d}^{3}r\Big{(}\hat{\psi}^{\dagger}_{a}\Big{[}-\frac{\hbar^{2}}{2}\nabla^{2}+V_{a}(\vec{r})+\frac{g_{a}}{2}\hat{\psi}^{\dagger}_{a}\hat{\psi}_{a}\Big{]}\hat{\psi}_{a}$
(2) $\displaystyle+$
$\displaystyle\hat{\psi}^{\dagger}_{m}\Big{[}-\frac{\hbar^{2}}{4}\nabla^{2}+V_{m}(\vec{r})+\epsilon+\frac{g_{m}}{2}\hat{\psi}^{\dagger}_{m}\hat{\psi}_{m}\Big{]}\hat{\psi}_{m}$
$\displaystyle+$ $\displaystyle
g_{am}\hat{\psi}^{\dagger}_{a}\hat{\psi}_{a}\hat{\psi}^{\dagger}_{m}\hat{\psi}_{m}+\frac{\alpha}{\sqrt{2}}\big{[}\hat{\psi}^{\dagger}_{m}\hat{\psi}_{a}\hat{\psi}_{a}+\hat{\psi}_{m}\hat{\psi}^{\dagger}_{a}\hat{\psi}^{\dagger}_{a}\big{]}\Big{)}$
(3)

Here, $g_{a}$, $g_{m}$, and $g_{am}$ quantify the strength of atom-atom,
molecule-molecule, and atom-molecule interactions, respectively. The trapping
potentials for atoms and molecules are symbolized as $V_{a}$ and $V_{m}$,
while $\alpha$ denotes the strength of photoassociation (PA). The parameter
$\epsilon$ accounts for the energy mismatch during the conversion of atoms to
molecules.

In the following, we focus on a cigar-shaped geometry. We maintain the
modified parameters from the quasi-one-dimensional geometry [32, 33] for ease
of notation. The equations of motion for the atomic and molecular mean fields
are expressed as:

$\displaystyle i\frac{\partial\psi_{a}}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\partial^{2}\psi_{a}}{\partial
x^{2}}+(V_{a}+g_{a}|\psi_{a}|^{2}+g_{am}|\psi_{m}|^{2})\psi_{a}$
$\displaystyle+$ $\displaystyle\alpha\sqrt{2}\psi_{m}\psi_{a}^{*},$
$\displaystyle i\frac{\partial\psi_{m}}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\frac{\partial^{2}\psi_{m}}{\partial
x^{2}}+(V_{m}+\epsilon+g_{m}|\psi_{m}|^{2}+g_{am}|\psi_{a}|^{2})\psi_{m}$
$\displaystyle+$ $\displaystyle\frac{\alpha}{\sqrt{2}}\psi_{a}^{2}.$

Clearly, when $\alpha$ is non-zero, the total particle count remains
conserved,

$N=\int{(|\psi_{a}|^{2}+2|\psi_{m}|^{2}})dx=N_{a}+2N_{m}.$ (6)

Here $\psi_{j}$’s are taken as,
$\psi_{j}(x,t)=\sqrt{n_{j}(x,t)}e^{i\phi_{j}(x,t)}$ for $j=a,m$., for which
the continuity equation can be written as,

$\frac{\partial}{\partial{t}}(n_{a}+2n_{m})+\frac{\partial}{\partial{x}}(\sum_{a,m}n_{j}\frac{\partial\phi_{j}}{\partial
x})=0.$ (7)

##  3 Quantum droplets

The coupled AMBEC system, being in general non-integrable, is not amenable to
a systematic approach for finding the solution space, unlike the NLSE, where
the hierarchy of solutions exists. It is important to note that for the same
molecular condensate wave function, multiple atomic wave functions can exist
in different parameter domains. Recently extended soliton’s solution has been
identified [34, 35]. Their collisional behaviour has been investigated by P.
Das et al.[36]. Here, we demonstrate the realization of atomic qubits with the
following three analytic solutions. The allowed parameter domain and
parametric conditions are given in the Supplementary material. These solutions
are not only characteristically different but will also have different atom
numbers. Physically, the droplets may be thought of as superposed solutions,
as will be seen.

We now show that these coupled equations given in Eq. 2 and 2 with the
condition $V_{1,2}(x)=0$ admit three classes of novel solutions, where for the
same $\psi_{m}$, we obtain two solutions in $\psi_{a}$, one corresponding to
the ground state and the other corresponding to the first excited state.

###  3.1  Solution I: Ground state solution in $\psi_{a}$

The exact solution to the coupled Eqs. (2) and (2) has the following form

$\psi_{a}=\frac{A\cosh(\beta x)}{B+\cosh^{2}(\beta x)}e^{-i\mu t}\,,$ (8)
$\psi_{m}=\frac{D}{B+\cosh^{2}(\beta x)}e^{-2i\mu t}\,,$ (9)

provided the desired consistency conditions are satisfied. These are depicted
in the Supplementary material. It is to be noted that $\psi_{a}$ and
$\psi_{m}$, the spatial wavefunction can be expressed as the following
superposition [37]:

$\frac{1}{B+\cosh^{2}(\beta x)}=\frac{\tanh(\beta x+\Delta)-\tanh(\beta
x-\Delta)}{\sinh(2\Delta)},$ $\frac{\cosh(\beta x)}{B+\cosh^{2}(\beta
x)}=\frac{\sech(\beta x+\Delta)+\sech(\beta x-\Delta)}{2\cosh(\Delta)}$

, where $B=\sinh^{2}(\Delta)$. Physically, this indicates the droplet can be
interpreted as a combination of kink and anti-kink, whereas the atomic profile
is a superposition of bright solitons.

The solutions can be cast in a form which explicitly shows the range of
chemical potential in which the solutions manifest [11]. With
$\beta^{2}=-2\mu$, we can reexpress the solution given by Eqs. (8) and (9) as

$\psi_{a}=\frac{2A\cosh(\sqrt{-2\mu}x)}{(2B+1)[1+\sqrt{1-\frac{\mu}{\mu_{0}}}\cosh(\sqrt{-8\mu}x)]}e^{-i\mu
t}\,,$ (10)
$\psi_{m}=\frac{\sqrt{n_{m}}\frac{\mu}{\mu_{0}}}{[1+\sqrt{1-\frac{\mu}{\mu_{0}}}\cosh(\sqrt{-8\mu}x)]}e^{-2i\mu
t}\,,$ (11)

where

$\sqrt{n_{m}}=\frac{D(2B+1)}{2B(B+1)}\,,~{}~{}\frac{\mu}{\mu_{0}}=\frac{4B(B+1)}{(2B+1)^{2}}\,.$
(12)

###  3.2 Solution II: First excited state solution in $\psi_{a}$

The excited atomic state can also be obtained exactly, which has the following
form

$\psi_{a}=\frac{A\sinh(\beta x)}{B+\cosh^{2}(\beta x)}e^{-i\mu t}\,,$ (13)
$\psi_{m}=\frac{D}{B+\cosh^{2}(\beta x)}e^{-2i\mu t}\,,$ (14)

provided certain relations are satisfied, which are given in Supplementary
material. We note that the atomic excited state profile can also be written in
the form

$\frac{\sinh(\beta x)}{B+\cosh^{2}(\beta x)}=\frac{\sech(\beta
x-\Delta)-\sech(\beta x+\Delta)}{2\sinh(\Delta)}$

with $B=\sinh^{2}(\Delta)$, which shows that the superposition of bright
solitons with opposite parity leads to the excited state.

Using $\beta^{2}=-2\mu$ , we re-express the solution given by Eqs. (13) and
(14) as

$\psi_{a}=\frac{2A\sinh(\sqrt{-2\mu}x)}{(2B+1)[1+\sqrt{1-\frac{\mu}{\mu_{0}}}\cosh(\sqrt{-8\mu}x)]}e^{-i\mu
t}\,,$ (15)
$\psi_{m}=\frac{\sqrt{n_{m}}\frac{\mu}{\mu_{0}}}{[1+\sqrt{1-\frac{\mu}{\mu_{0}}}\cosh(\sqrt{-8\mu}x)]}e^{-2i\mu
t}\,,$ (16)

where

$\sqrt{n_{m}}=\frac{D(2B+1)}{2B(B+1)}\,,~{}~{}\frac{\mu}{\mu_{0}}=\frac{4B(B+1)}{(2B+1)^{2}}\,.$
(17)

For both ground state and excited state, the molecular BEC solution,
$\psi_{m}$, has the same form, and $\psi_{m}$ becomes constant as
$\mu\rightarrow\mu_{0}$.

##  4 Pulse Solution with Power Law Tail

###  4.1 Solution III: Ground state solution in $\psi_{a}$

There is another class of solution which shows pulse behaviour. In particular,
we can check the exact pulse solution to the coupled Eqs. (2) and (2) is given
by

$\psi_{a}=\frac{A}{B+x^{2}}e^{-i\mu t}\,,$ (18)
$\psi_{m}=\frac{D(x^{2}+y)}{B+x^{2}}e^{-2i\mu t}\,,$ (19)

where $y$ is any real number with $y\neq B$, provided some consistency
conditions given in Supplementary material are satisfied.

###  4.2 Solution IV: First excited state solution in $\psi_{a}$

It is easy to check that another exact solution to the coupled Eqs. (2) and
(2) is

$\psi_{a}=\frac{Ax}{B+x^{2}}e^{-i\mu t}\,,$ (20)
$\psi_{m}=\frac{D(x^{2}+y)}{B+x^{2}}e^{-2i\mu t}\,,$ (21)

where $y$ is any real number with $y\neq B$, provided the consistency
conditions given in the Supplementary material are satisfied.

##  5 Hyperbolic Solution

###  5.1 Solution V: Ground State Solution in $\psi_{a}$

We find an exact hyperbolic solution to the coupled equations (2) and (2),
which is

$\psi_{a}=A\sech^{2}(\beta x)e^{-i\mu t}\,,$ (22) $\psi_{m}=B[\sech^{2}(\beta
x)+y]e^{-2i\mu t}\,,$ (23)

provided the relations given in the Supplementary material are satisfied. We
have two classes of constraints depending on whether $y=0$ or $y\neq 0$.

###  5.2 Solution VI: First Excited State Solution in $\psi_{a}$

We find exact hyperbolic solution to the coupled Eqs.(2) and ((2)) having the
form

$\psi_{a}=A\sech(\beta x)\tanh(\beta x)e^{-i\mu t}\,,$ (24)
$\psi_{m}=B[\sech^{2}(\beta x)+y]e^{-2i\mu t}\,,$ (25)

provided some relations are satisfied, given in Supplementary material.
Depending on whether $y=0$ or $y\neq 0$, two classes of constraints are
obtained.

##  6 Realization of Quantum Gates

One-qubit gates are particular unitary transformations represented by 2x2
complex matrices designed to change the state of a single qubit. Examples of
such gates include the NOT gate, which flips the state $|0\rangle$ to
$|1\rangle$ and vice versa, and the Hadamard (H) gate, which transforms a pure
state into a superposition of both $|0\rangle$ and $|1\rangle$. In the basis
$\\{|0\rangle,|1\rangle\\}$, these gates are represented as follows:

$NOT=\begin{bmatrix}0&1\\\
1&0\end{bmatrix},H=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix}$

The superposition of macroscopic Bose-Einstein condensate states has been
shown both theoretically and experimentally. In Ref. [38], authors have shown
instead of analyzing spatial interference patterns, one can combine scattered
photons from atomic transitions between different BECs using a photon beam
splitter. Simulations show that detecting scattered photons drives the
condensates into macroscopic quantum superpositions of phase and number state.
This method is theoretically advantageous over atom counting and
nondestructive to condensates. In Ref. [39], it has been shown that matter
wave interference patterns can be observed in the intra-trap collision of two
bright solitons by selectively tuning the trap frequency and scattering
length. The ground and excited atomic states of the AMBEC in the present case
differ by parity. We observe that for creating superposition of the atomic
molecular BEC states, the angular momentum carrying Lauggere Gaussian beams
can be potentially useful [40]. These techniques can be used to make quantum
superpositions of BEC, which eventually become the operation of the Hadamard
gate. Nevertheless, creating the superposition of these atomic states needs
careful consideration.

The other important CNOT gate for carrying out the entangling operation alters
the state of a “target” qubit only when a “control” qubit is in the state
$\ket{1}$. In the context of BEC, Al Khawaja et al. introduce a protocol for
the quantum controlled-NOT gate, employing a reflectionless potential well in
an optical system to manipulate two qubits during soliton scattering [13]. It
is worth mentioning that in the context of optical fibre, soliton manipulation
for achieving gate operation has been realized. For the integrable Manakov
case, computational universality of the solitons has been established [12].
Similar methods can be explored in the present case.

##  7 Conclusion

In conclusion, we have explicitly constructed a number of two-level systems
with characteristically different macroscopic wavefunctions for the atomic
condensate in an atom-molecular BEC. These include hyperbolic $\sinh(\beta x)$
and $\cosh(\beta x)$ type atomic condensate wavefunctions in a molecular
droplet background having a flat top characteristic. These solutions are well-
controlled by the photo association and exist in a range of negative chemical
potentials. The dark and bright solitonic configurations are also shown to
provide a two-level system, similar to the solitons of the Manakov system.
Interestingly, in one case, the atomic qubits are produced by the
superposition of kink anti-kinks and bright solitons with appropriate parity.
In another realization, the atomic qubits are in the molecular form, being a
product of bight and kink solitons. The pulse-type solitons are also
identified as qubits, which have a Lorentzian density profile, showing power
law decay. Expectedly, these spatially extended solutions are well-localized
in the momentum domain, providing a characteristically different class of
qubit state. We have also discussed the procedure for creating energy level
superposition to realize the Hadamard gate, as well as the possible
realization of the CNOT gate. The detailed investigation of gate operations,
as well as scalability [41], are under investigation.

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