# Dynamical galactic effects induced by stable vortex structure in bosonic
dark matter

K. Korshynska  Department of Physics, Taras Shevchenko National University of
Kyiv, 64/13, Volodymyrska Street, Kyiv 01601, Ukraine  Physikalisch-Technische
Bundesanstalt (PTB), Bundesallee 100, D-38116 Braunschweig, Germany  Y.M.
Bidasyuk  Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100,
D-38116 Braunschweig, Germany  E.V. Gorbar  Department of Physics, Taras
Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, Kyiv
01601, Ukraine  Bogolyubov Institute for Theoretical Physics, 14-b
Metrolohichna Street, Kyiv 03143, Ukraine  Junji Jia  School of Physics and
Technology, Wuhan University, 299 Bayi Roadd, Wuhan, Hubei Prov., China 430072
A.I. Yakimenko  Department of Physics, Taras Shevchenko National University of
Kyiv, 64/13, Volodymyrska Street, Kyiv 01601, Ukraine  Dipartimento di Fisica
e Astronomia ’Galileo Galilei’, Universit‘a di Padova, via Marzolo 8, 35131
Padova, Italy  Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via
Marzolo 8, 35131 Padova, Italy

###### Abstract

The nature of dark matter (DM) remains one of the unsolved mysteries of modern
physics. An intriguing possibility is to assume that DM consists of ultralight
bosonic particles in the Bose-Einstein condensate (BEC) state. We study
stationary DM structures by using the system of the Gross-Pitaevskii and
Poisson equations, including the effective temperature effect with parameters
chosen to describe the Milky Way galaxy. We have investigated DM structure
with BEC core and isothermal envelope. We compare the spherically symmetric
and vortex core states, which allows us to analyze the impact of the core
vorticity on the halo density, velocity distribution, and, therefore, its
gravitational field. Gravitational field calculation is done in the
gravitoelectromagnetism approach to include the impact of the core rotation,
which induces a gravimagnetic field. As result, the halo with a vortex core is
characterized by smaller orbital velocity in the galactic disk region in
comparison with the non-rotating halo. It is found that the core vorticity
produces gravimagnetic perturbation of celestial body dynamics, which can
modify the circular trajectories.

##  I Introduction

The nature of DM particles remains one of the most fascinating puzzles of
modern physics. The DM large-scale properties consistent with astrophysical
observations are successfully explained by the cold dark matter model (CDM),
which describes DM as a collisionless sufficiently cold perfect fluid.
However, at smaller scales, the CDM encounters the cusp-core, missing
satellites, and too-big-to-fail problems. One possibility to solve them is to
assume that DM particles are ultra-light bosons as it is assumed in ultra-
light dark matter (ULDM) models [1]. Generically, these models are
characterized by the suppression of the small-scale structures, the presence
of cores, and dynamic effects which arise from the BEC formed in the central
regions of galaxies. Such DM halo proposals were investigated in [2, 3, 4, 5,
6].

The ULDM model is supported indirectly by observations. For example, in
cosmological simulations [7] it was found that the bosonic DM can indeed
reproduce the observed distribution of matter at very large scales [8, 9],
though the mass of such bosons should be extremely small. There have been also
studies on some tensions of the ULDM with observational data from the rotation
curves of galaxies including the Milky Way, which could probe the particle
mass in the range $m=10^{-22}-10^{-21}$ eV [10, 11]. Furthermore, the
viability of the ULDM model was studied with the stellar kinematics
measurements in dwarf galaxies [12]. Another important piece of evidence is
the DM nongravitational self-interaction, which has been recently reported for
collisions of the clusters [13, 14]. In addition, the DM halo model must
ensure the stability of a predicted halo. The stability of compact
astrophysical objects which may be formed due to the Bose-Einstein
condensation of ULDM was shown numerically [15].

In the present paper, we discuss DM, which consists of ultra-light bosons with
repulsive self-interaction. Such models make use of two macroscopic quantum
phenomena: Bose-Einstein condensation and superfluidity. Bose-Einstein
condensate in the mean-field approximation is described by the Gross-
Pitaevskii equation. By adding dissipation in the Gross-Pitaevskii equation
one obtains a more general model, which includes the effective temperature
effect and predicts that the ULDM halo consists of a BEC core and an
isothermal envelope [16]. Such core-envelope structure in the ULDM model was
also discussed in [17, 18, 19, 20]. Another important property, superfluidity,
allows the quantization of the circulation and thus the possibility of the
formation of vortices in the core of the halo. The central object of our
study, the vortex, has a vanishing wavefunction at the vortex line, with a
quantized circular flow around the vortex line [1]. According to the recent
numerical studies [21, 22] only the non-rotating soliton and single-charged
vortex are stable, even being strongly perturbed. In the present work, we
consider a DM halo, which consists of two regions - core and isothermal
envelope, while the core could be either a soliton or a single-charged vortex.

Most of our knowledge about DM is based on its gravitational interaction with
baryonic matter objects. Thus, testing the validity of the UDM theory requires
a detailed investigation of the DM gravitational field. The DM density
distribution, predicted by ULDM models, has been extensively studied in
numerical simulations and applied in studies aimed at reconstructing the
gravitational potential of DM halos for the Milky Way [23] and dwarf galaxies
[24]. In general, one can determine the gravitational field of the ULDM by
solving the Einstein equations with the DM density and rotation flow as
sources of the gravitational field, where rotation flow is induced by the BEC
superfluidity. Thus, in the ULDM model, we should be able to deduce the impact
of the superfluid DM rotation on the observations. The dominant effect of the
vortex existence is due to the different core density distributions. Moreover,
rotation flows produce $v/c$ and higher order effects, which can be taken into
account in the gravitoelectromagnetism approach discussed in [25, 26, 27, 28,
29] and used in our calculations below. The gravitoelectromagnetic formulation
of a slowly rotating, self-gravitating, and dilute BEC intended for
astrophysical applications in the context of DM halos was discussed in [30].
As a rule, the gravimagnetic force is quite weak and does not affect
significantly the dynamics of astrophysical systems. However, in the central
region of the BEC core, the DM density vanishes while the vortex flow velocity
dramatically increases, which can affect the dynamics of luminous matter in
the central region of the galaxies.

In the present work, we calculate the DM gravitational field, which is needed
for analysis of the observable predictions of the DM model, namely, to study
how DM affects the movement of luminous matter. In our study, DM is the only
source of a gravitational field, while luminous matter moves along geodesics,
induced by DM. A more precise description of galactic kinematics is given by
modeling the baryonic contribution to the gravitational potential which can
distort the BEC soliton structures [31, 32]. Such a contribution was found to
be significant for the Milky Way (MW) but not essential for the SPARC LSB
galaxies [33]. In this paper, we will limit ourselves to some simple
consequences of the ULDM model on the galactic kinematics, namely, rotation
curves and deviation of circular trajectory, induced by the gravimagnetic
force. The more detailed study in this direction is beyond the scope of the
current paper, though it is an interesting perspective on further work.

The paper is organized as follows. In Sec.II, we develop the key parameters of
our model, define the equations for halo structure, and formulate the
gravitoelectromagnetism ansatz. In Sec.III, we discuss the halo density
profile for two stable core configurations and define the corresponding
hydrodynamical velocity. In Sec.IV, the gravielectric (Newtonian) field of the
halo is calculated and the rotational curves are obtained. Sec.V provides
gravimagnetic field calculations and our estimates of the gravimagnetic effect
on circular trajectory. The results are summarized in Sec.VI.

##  II Model

###  II.1 Ultra-light dark matter model and halo structure

In this section, we briefly discuss the model, suggested in [16]. The
structure of the DM halo is described by the Gross-Pitaevskii-Poisson (GPP)
equations, which define the dynamical evolution of self-gravitating BEC field
$\psi$

$\displaystyle i\hbar\frac{\partial\psi}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\Delta\psi+m\Phi_{\mathrm{g}}\psi+\frac{K\gamma
m}{\gamma-1}|\psi|^{2(\gamma-1)}\psi$ (1)
$\displaystyle+\frac{m}{2}\left(\frac{3}{4\pi\eta_{\mathrm{0}}}\right)^{2/3}|\psi|^{4/3}\psi+2k_{\mathrm{B}}T\ln|\psi|\psi$
$\displaystyle-i\frac{\hbar}{2}\xi\left[\ln\left(\frac{\psi}{\psi^{*}}\right)-\left\langle\ln\left(\frac{\psi}{\psi^{*}}\right)\right\rangle\right]\psi,$
$\Delta\Phi_{\mathrm{g}}=4\pi G|\psi|^{2},$ (2)

where $\langle X\rangle=\frac{1}{M}\int|\psi|^{2}Xd\mathbf{r}$ is the spatial
average over halo, $m$ is the bosonic particle mass, $\hbar$ denotes reduced
Planck constant, $k_{\mathrm{B}}$ is Boltzmann constant. The first equation
can be obtained by incorporating dissipative effects into the Schrödinger
equation by means of the theory of scale relativity. This generalization of
the Schrödinger equation means basically taking into account the interaction
of the system with the external environment. The model Eqs. (1),(2) were
derived in [34], and we follow this approach in our current work.

We consider the BEC model with parameters $\gamma=2$ and $K=\frac{2\pi
a_{\mathrm{s}}\hbar^{2}}{m^{3}}$, where $a_{\mathrm{s}}$ denotes the $s$-wave
scattering length of the self-interaction. The parameter $\eta_{\mathrm{0}}$
determines the equation of state of DM [35]. The first term on the right-hand
side of Eq.(1) is the kinetic term, and the second describes the interaction
with the condensate gravitational potential $\Phi_{\mathrm{g}}$. The third
term takes into account the bosonic self-interaction (we will consider only
the case $\gamma=2$ which corresponds to binary collisions). The fourth term
accounts for the core, and the fifth term describes an isothermal envelope
with effective temperature $T$ which surrounds the core. These terms can be
derived from the Lynden-Bell theory of violent relaxation [16]. The last term
with $\xi<0$ is a damping term and ensures that the system relaxes towards the
equilibrium state.

An important feature of the Gross-Pitaevskii (GP) equation is that it
satisfies the H-theorem, i.e., the free energy $F$ of the system decreases

$\dot{F}=-\xi\int\rho\mathbf{u}^{2}d\mathbf{r}\leq 0.$

where $\rho=|\psi|^{2}$ denotes BEC density and $\mathbf{u}=\nabla
S(\mathbf{r},t)/m$ is the velocity field. These quantities are obtained by
application of Madelung transformation, according to the expression
$\psi(\mathbf{r},t)=\sqrt{\rho(\mathbf{r},t)}e^{iS(\mathbf{r},t)/\hbar}$,
where $S(\mathbf{r},t)$ is the action. The negative sign of $\xi$ implies that
the system relaxes towards the state with zero hydrodynamical velocity
$\mathbf{u}=0$. Therefore, a stationary vortex solution with nonzero
$\mathbf{u}$ can be found only if we set $\xi=0$.

The free energy $F=E-TS_{\mathrm{B}}$ is expressed through the total energy
$E$, the effective temperature $T$, and the Boltzmann entropy
$S_{\mathrm{B}}=-k_{\mathrm{B}}\int(\rho/m)(\ln\rho-1)d\mathbf{r}$. The total
energy consists of the classical kinetic energy
$\Theta_{\mathrm{c}}=1/2\int\rho\mathbf{u}^{2}d\mathbf{r}$, the quantum
kinetic energy $\Theta_{\mathrm{Q}}=1/m\int\rho Qd\mathbf{r}$, the
gravitational potential energy $W=1/2\int\rho\Phi_{\mathrm{g}}d\mathbf{r}$,
and the internal energy of the self-interaction $U=K\int\rho^{2}d\mathbf{r}$,
$E_{\mathrm{0}}=\Theta_{\mathrm{c}}+\Theta_{\mathrm{Q}}+W+U$. Here
$Q=-\frac{\hbar^{2}}{2m}\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}$ is the quantum
potential. A stable equilibrium state corresponds to the minimum of the free
energy $F$ at fixed total mass $M$ of BEC. This gives the following condition
of quantum hydrostatic equilibrium [16]:

$\frac{\rho}{m}\nabla Q+\nabla
P+\rho\nabla\Phi_{\mathrm{g}}+\frac{\rho}{2}\nabla\mathbf{u}^{2}=0,$

where $P=K\rho^{2}+\rho\frac{k_{B}T}{m}$ is pressure due to the self-
interaction and effective temperature. Taking into account the Poisson
equation (2) and neglecting the quantum pressure term $Q$, we obtain the
following equation of state

$-2K\Delta\rho-\frac{k_{\mathrm{B}}T}{m}\Delta\ln\rho=4\pi
G\rho+\frac{1}{2}\mathbf{u}^{2},$ (3)

where $G$ is the gravitational constant. The solution of this equation is
discussed in Sec. III.

###  II.2 Gravitoelectromagnetic approach

To determine the gravitational field of DM halo we employ the well-known
gravitoelectromagnetism (GEM) approach [29] which was previously applied to
galactic structures in [25, 27, 28]. According to the GEM formalism, in the
case of a test particle (which is luminous matter in our case) moving much
slower than the speed of light $c$, it is convenient to represent the
spacetime metric in the form

$dS^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=\left(1-\frac{2\Phi_{\mathrm{g}}}{c^{2}}\right)(dx^{0})^{2}\\\
+\frac{4}{c^{2}}\left(\mathbf{A}_{\mathrm{g}}\mathbf{dx}\right)dx^{0}+\left(-1-\frac{2\Phi_{\mathrm{g}}}{c^{2}}\right)\delta_{ij}dx^{i}dx^{j},$
(4)

where $\Phi_{\mathrm{g}}$ and $\mathbf{A}_{\mathrm{g}}$ are the GEM scalar
(gravielectric) and vector (gravimagnetic) potentials. For the
gravitoelectromagnetic fields $\mathbf{E}_{\mathrm{g}}$ and
$\mathbf{B}_{\mathrm{g}}$

$\mathbf{E}_{\mathrm{g}}=-\nabla{\Phi}-\frac{1}{2c}\partial_{t}\mathbf{A}_{\mathrm{g}},$
(5) $\mathbf{B}_{\mathrm{g}}=\nabla\times\mathbf{A}_{\mathrm{g}},$ (6)

the Einstein equations imply the following relations:

$\nabla\mathbf{E}_{\mathrm{g}}=4\pi G\rho,$
$\nabla\times\mathbf{B}_{\mathrm{g}}=\frac{2}{c}\partial_{t}\mathbf{E}_{\mathrm{g}}+\frac{8\pi
G}{c}\mathbf{j}.$

Here sources of the gravitational field are mass density $\rho$ and matter
current $\mathbf{j}=\rho\mathbf{u}$ ($\mathbf{u}$ is the matter velocity).

Since these equations are clearly analogous to those in the electromagnetic
theory, their solutions have a form similar to Maxwell’s theory

$\Phi_{\mathrm{g}}(\mathbf{r})=G\int_{\Omega}\frac{\rho(\mathbf{r^{\prime}})d^{3}r^{\prime}}{|\mathbf{r}-\mathbf{r^{\prime}}|},$
(7)
$\mathbf{A_{\mathrm{g}}}(\mathbf{r})=\frac{2G}{c}\int_{\Omega}\frac{\rho(\mathbf{r^{\prime}})\mathbf{u}(\mathbf{r^{\prime}})d^{3}r^{\prime}}{|\mathbf{r}-\mathbf{r^{\prime}}|},$
(8)

where integration proceeds over $\mathbf{r^{\prime}}$ occupied by DM
particles, $\rho(\mathbf{r^{\prime}})$ is the condensate density and
$\mathbf{u}(\mathbf{r^{\prime}})$ is the BEC velocity at
$\mathbf{r^{\prime}}$. $\mathbf{r}$ are coordinates associated with the test
particle, moving along geodesics in the BEC gravitational field.

Finally, the geodesic movement for a test particle, which corresponds to the
spacetime metric in the GEM form,

$\frac{d^{2}x^{i}}{dt^{2}}=\frac{\partial\Phi_{\mathrm{g}}}{\partial
x_{i}}+\frac{2}{c}\frac{dA_{\mathrm{g}}^{i}}{dt}-\frac{2}{c}\left(\frac{\partial\mathbf{A}_{\mathrm{g}}}{\partial
x_{i}}\frac{d\mathbf{x}}{dt}\right)$

can be equivalently described as the classical motion
$m\ddot{\mathbf{x}}=\mathbf{F}_{\mathrm{g}}$ in the gravitoelectromagnetic
analog of the Lorentz force

$\mathbf{F}_{\mathrm{g}}=-m\left(\mathbf{E}_{\mathrm{g}}+\frac{2}{c}\mathbf{v}\times\mathbf{B}_{\mathrm{g}}\right)=m(\mathbf{a}_{\mathrm{E}}+\mathbf{a}_{\mathrm{B}}),$
(9)

where $\mathbf{v}$ is the particle velocity and $m$ is its mass. Here we
introduced gravielectric $\mathbf{a}_{\mathrm{E}}=-\mathbf{E}_{\mathrm{g}}$
and gravimagnetic
$\mathbf{a}_{\mathrm{B}}=-\frac{2}{c}\mathbf{v}\times\mathbf{B}_{\mathrm{g}}$
components of acceleration.

##  III Halo density profile

The model, based on the generalized GPP equations (see Eqs. (1), (2))
describes the core-envelope structure of DM halo with a dense core and diffuse
isothermal envelope. The model yields the following equation of state for the
ULDM $P=K\rho^{2}+\rho\frac{k_{B}T}{m}$ (see Sec. II). Thus, one can conclude,
that in the core region equation of state is approximately $P=K\rho^{2}$,
because the weak self-interaction dominates over effective temperature impact
due to large density. That is why the latter will be neglected in the
discussion of the core states. On the contrary, in the isothermal envelope
region, we have the equation of state $P=\rho\frac{k_{B}T}{m}$, which means
that the effective temperature term plays a crucial role there.

Based on these considerations, we calculate the halo density in two steps.
Firstly, we reproduce the numerical result for the total density of the non-
rotating halo (see the original result in [16]), which defines density
distribution in the isothermal envelope region. This step is needed as a
starting point to define isothermal envelope density distribution and to
compare the discussed in [16] $s=0$ solitonic core with the new case of vortex
core $s=1$. Secondly, under the assumption that core and envelope do not
interact, we discuss the core density profile separately by means of
variational ansatz [36]. This way we will study the spherically symmetric
($s=0$) and the single-charged vortex ($s=1$) solutions for the core density
distribution.

###  III.1 Isothermal envelope

In the first case of a non-rotating core, we can set $\mathbf{u}=0$ , and then
the Eq. (3) simplifies

$-\frac{4\pi
a_{\mathrm{s}}\hbar^{2}}{m^{3}}\Delta\rho-\frac{k_{\mathrm{B}}T}{m}\Delta\ln\rho=4\pi
G\rho,$

where we took into account that $K=\frac{4\pi
a_{\mathrm{s}}\hbar^{2}}{m^{3}}$.

It is convenient to introduce the density function and the radial coordinate
$\rho=\rho_{\mathrm{c}}e^{-f}$, $y=r/r_{\mathrm{0}}$, where

$r_{\mathrm{0}}=\sqrt{\frac{k_{\mathrm{B}}T}{4\pi G\rho_{\mathrm{c}}m}}$ (10)

and $\rho_{\mathrm{c}}$ defines the density at the center. The equation of
state can be rewritten in the following form:

$\frac{d^{2}f}{dy^{2}}+\frac{2}{y}\frac{df}{dy}=\frac{\chi\left(\frac{df}{dy}\right)^{2}+1}{\chi+e^{f}},$
(11)

where $\chi=4\pi
a_{\mathrm{s}}\hbar^{2}\rho_{\mathrm{c}}/(m^{2}k_{\mathrm{B}}T)$. The boundary
conditions are $f(0)=0$ and $\frac{df}{dy}(0)=0$, which define the boundary
conditions for the density function $\rho(0)=\rho_{\mathrm{c}}$ and
$\frac{d\rho(0)}{dr}=0$. We solve Eq. (11) numerically for different values of
$\chi$ and present solutions in Fig. 1 (a). The isothermal envelope density
distribution is defined as $\rho=\rho_{0}e^{-f}=\rho_{0}f_{\mathrm{N}}(r)$,
where $f$ is a numerical solution of Eq. (11).

The profile has a solitonic core and an isothermal envelope whose density
decreases as $\rho(r)\sim k_{\mathrm{B}}T/(2\pi Gmr^{2})=v_{\infty}^{2}/(4\pi
Gr^{2})$ [16] in agreement with observations (here $v_{\infty}$ is the
constant rotational velocity in the large distance limit). The existence of a
BEC core in the ULDM model was also discussed in [17, 18, 19, 20].

The possible physical origin of the core-envelope structure could be the
merger of two-state configurations when the total system tends to a virialized
state, and the obtained averaged profile has a core and a tail structure [37].
The process of halo formation usually undergoes gravitational cooling [38],
which is discussed in [39, 38]. Gravitational cooling process for inreitially
quite arbitrary density profiles leads to relaxation and virialization through
the emission of scalar field particles [40]. The resulting profile has the
same dense core and diffuse envelope structure.

In the case $s=1$, the hydrodynamical velocity $\mathbf{u}$ does not vanish in
the inner region due to the existence of the vortex. The definition of the
velocity profile in the isothermal halo region is a complicated task. One
would expect that there is an intermediate region between the core and
isothermal envelope, where the hydrodynamical velocity is small but nonzero,
and at large enough distances, we should have $\mathbf{u}=0$. This is due to
the divergent mass of the isothermal envelope, which therefore cannot rotate
in order for kinetic energy to be finite. For an estimate, we simply put
$\mathbf{u}=0$ in the whole isothermal envelope region. This approximation can
be justified by the negligibly small density of the isothermal envelope in
comparison with the core density, so its rotation would have no sufficient
impact on the system. Hence the density profile in the envelope region remains
unchanged. Thus, to define isothermal envelope density distribution we use the
numerical solution for $\rho=\rho_{0}f_{\mathrm{N}}(r)$, obtained earlier in
the case of non-rotating core. The density profile in the core region will be
discussed in the next section in detail.

To reproduce the Milky Way halo mass $M=1.3\times 10^{12}M_{\odot}$ and radius
$R_{\mathrm{halo}}=287kpc$ [41], taking into account the model described in
[16], we choose the following values of the particle mass $m=2.92\times
10^{-22}eV/c^{2}=0.52\times 10^{-57}kg$, scattering length
$a_{\mathrm{s}}=8.17\times 10^{-77}m$, effective DM temperature $T=5.09\times
10^{-25}K$, central density in the spherical case
$\rho_{\mathrm{c}}=0.34\times 10^{-17}\frac{kg}{m^{3}}$ and distance scaling
parameter $r_{\mathrm{0}}=0.071kpc$. Then $\chi=20$ and the temperature of BEC
of such ultralight bosons is much larger than the effective temperature. For
the spherically symmetric case, this yields the core with mass
$M_{\mathrm{c}}=6.39\times 10^{10}M_{\odot}$ and radius $R_{\mathrm{c}}=1kpc$.

###  III.2 Core stationary states

The dynamics of self-gravitating BEC of $N$ weakly interacting bosons with
mass $m$ is described by the GPP system of equations with the term,
corresponding to the effective temperature impact:

$i\hbar\frac{\partial\psi}{\partial
t}=\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+gN|\psi|^{2}+m\Phi_{g}\right.\\\
\left.+2k_{B}T\ln\Big{|}\frac{\psi}{\psi_{0}}\Big{|}\right)\psi$ (12)
$\nabla^{2}\Phi_{g}=4\pi GmN|\psi|^{2}$ (13)

where $g=\frac{4\pi\hbar^{2}a_{s}}{m}$ is the coupling strength that
corresponds to the two-particle interaction, $a_{s}$ is the s-wave scattering
length, $\Phi$ is the gravitational potential and G is gravitational constant.

The GPP system of Eqs.(12) and (13) includes three crucial physical
parameters: particle mass $m$, the total number of particles $N$ (or,
equivalently, total mass $M$) and coupling strength $g$ (or, equivalently,
self-interaction constant
$\frac{\lambda}{8\pi}=\frac{a_{\mathrm{s}}}{\lambda_{\mathrm{c}}}$, where
$\lambda_{\mathrm{c}}=\frac{\hbar}{mc}$ is the Compton wavelength of bosons)
[36].

The GPP system of equations is invariant under the transformation
$t=\lambda_{*}^{2}t^{\prime}$, $\mathbf{r}=\lambda_{*}\mathbf{r}^{\prime}$,
$\psi=\lambda_{*}^{-2}\psi^{\prime}$,
$\Phi_{\mathrm{g}}=\lambda_{*}^{-2}\Phi^{\prime}_{\mathrm{g}}$,
$g=\lambda_{*}^{2}g^{\prime}$, where $\lambda_{*}>0$, which allows us to
scale-out the coupling constant to $g=1$.

In order to simplify calculations, it is convenient to introduce dimensionless
variables and wave function

$i\frac{\partial\psi}{\partial
t}=\left(-\frac{1}{2}\nabla^{2}+|\psi|^{2}+\Phi_{\mathrm{g}}+T_{\mathrm{eff}}\ln|\psi|\right)\psi,$
(14) $\nabla^{2}\Phi_{\mathrm{g}}=|\psi|^{2},$ (15)

where the dimensional variables are related to the dimensionless ones as
follows: $\mathbf{r}=\mathbf{r}_{\mathrm{ph}}{L}$,
$t=\omega_{*}t_{\mathrm{ph}}$,
$\Phi_{\mathrm{g}}=\left(\frac{\lambda_{\mathrm{c}}}{L}\right)^{2}\frac{\Phi_{\mathrm{gph}}}{c^{2}}$,
and
$\psi=\frac{\lambda}{8\pi}\left(\frac{m_{\mathrm{Pl}}}{m}\right)^{2}\sqrt{4\pi
GM}\frac{\hbar}{mc^{2}}\psi_{\mathrm{ph}}$. Here the distance and time scaling
parameters are
$L=\lambda_{\mathrm{c}}\frac{m_{\mathrm{Pl}}}{m}\sqrt{\frac{\lambda}{8\pi}}=\frac{m_{\mathrm{Pl}}\hbar}{m^{2}c}\sqrt{\frac{\lambda}{8\pi}}=0.99\times
10^{19}m=0.32kpc$ and
$\omega_{*}=\frac{c\lambda_{\mathrm{c}}}{L^{2}}=2.08\times 10^{-15}s^{-1}$.
The dimensionless effective temperature parameter is
$T_{\mathrm{eff}}=\frac{2k_{\mathrm{B}}T}{\omega_{*}\hbar}$ and will be
neglected in the following discussion because the corresponding term
$T_{\mathrm{eff}}\ln|\psi|$ is negligibly small in the core region. Therefore,
we neglect the temperature effects in the analysis of the BEC core density
distribution .

For the BEC core mass $M_{\mathrm{c}}=6.39\times 10^{10}M_{\odot}$ and radius
$R_{\mathrm{c}}=1kpc$, we solve the GPP equations (14) and (15) by using the
variational ansatz in cylindrical coordinates $r,z$

$\psi(r,\phi,z)=A\left(\frac{r}{R}\right)^{s}e^{-\frac{r^{2}}{2R^{2}}-\frac{z^{2}}{2(R\eta)^{2}}+is\phi}.$
(16)

Here $R$ and $\eta$ are variational parameters, which will be fixed later.
Constant $A$ is fixed by the normalization condition

$A=\sqrt{\frac{N_{\mathrm{0}}}{\pi^{3/2}\eta R^{3}s!}},$ (17)

the cases $s=0,1$ are considered, and $N_{\mathrm{0}}$ is defined by the core
mass

$N_{\mathrm{0}}=4\pi\frac{M_{\mathrm{c}}}{m_{\mathrm{Pl}}}\sqrt{\frac{\lambda}{8\pi}}=2.55\cdot
10^{4},$

where $m_{\mathrm{Pl}}=\sqrt{\frac{\hbar c}{G}}$ is the Planck mass and
$\lambda/(8\pi)=1.21\times 10^{-91}$ is the self-interaction coupling
constant.

The dimensionless quantities and the physically observed ones are related as
follows:

$R_{\mathrm{c}}=R_{\mathrm{99}}L=\frac{m_{\mathrm{Pl}}\hbar}{m^{2}c}\sqrt{\frac{\lambda}{8\pi}}R_{\mathrm{99}},$
(18)
$\rho=M|\psi_{\mathrm{ph}}|^{2}=\frac{M}{L^{3}N_{\mathrm{0}}}|\psi|^{2}\\\
=\rho_{\mathrm{0}}\left(\frac{r}{R}\right)^{2s}e^{-\frac{r^{2}}{R^{2}}-\frac{z^{2}}{(R\eta)^{2}}},$
(19)

where $R_{\mathrm{99}}$ is the dimensionless radius which contains $99$
percent of the mass of the core (the variational analysis gives $R_{99}\approx
2.38R$ in the case of solitonic core and $R_{99}\approx 2.58R$ in the case of
vortex core), $\rho$ is the condensate density, and
$\rho_{\mathrm{0}}=MA^{2}/(L^{3}N_{\mathrm{0}})$ is the density scaling
parameter. $R_{\mathrm{c}}$ denotes the total radius of the core in physical
units.

Using the variational ansatz for the BEC wave function (16), we obtain the
energy [36]:

$E=\int
d^{3}\mathbf{r}\psi^{*}(\mathbf{r},t)\left(-\frac{1}{2}\nabla^{2}+|\psi|^{2}+\Phi_{\mathrm{g}}\right)\psi(\mathbf{r},t)\\\
=\epsilon\left(\frac{N_{\mathrm{0}}(1+2\eta^{2}(1+s))}{4R^{2}\eta^{2}}+\frac{N_{\mathrm{0}}^{2}\Gamma(s+1/2)}{4\sqrt{2}\pi^{2}R^{3}\eta\Gamma(s+1)}\right.\\\
-\left.\frac{N_{\mathrm{0}}^{2}}{8\pi
R}\int_{0}^{\infty}\operatorname{Erfc}\left(\frac{k_{*}\eta}{\sqrt{2}}\right)L_{\mathrm{s}}^{2}\left(\frac{k_{*}^{2}}{4}\right)e^{-\frac{k_{*}^{2}(1-\eta^{2})}{2}}dk_{*}\right).$
(20)

where $\Gamma(x)$ denotes the Gamma function, $\operatorname{Erfc}(x)$ is the
complementary error function and $L_{\mathrm{s}}(x)$ is the Laguerre
polynomials. Here $\epsilon=(\hbar^{2}/4\pi
m_{\mathrm{Pl}}\lambda_{\mathrm{c}}^{2})(8\pi/\lambda)^{3/2}$ is
characteristic energy, which does not depend on variational parameters.

In what follows, we will use $r_{\mathrm{0}}=2.18\times
10^{18}m=0.071kpc=0.22L$ as the distance scaling parameter.

In the subsection below, we investigate the case $s=0$.

####  III.2.1 Non-rotating spherically-symmetric core

In this case, the BEC wave function in Eq. (16) depends only on radial
distance $r$ in spherical coordinates

$\psi(r)=Ae^{-\frac{r^{2}}{2R^{2}}}$ (21)

and the density function (see Eq. (19)) equals

$\rho(r)=\rho_{\mathrm{0}}e^{-\frac{r^{2}}{R^{2}}}.$ (22)

In what follows, $r$ will denote spherical distance, when the $s=0$ case is
discussed.

We should relate $R$ and the BEC core radius $R_{\mathrm{c}}$ which is defined
through $M_{\mathrm{c}}=\frac{4}{\pi}\rho_{\mathrm{0}}R_{\mathrm{c}}^{3}$
[36]. Since $\rho_{\mathrm{0}}=M_{\mathrm{c}}A^{2}/(L^{3}N_{\mathrm{0}})$, the
numerical result for halo density (see Fig. 1 a) gives
$\frac{R_{\mathrm{c}}}{LR}=1.64$ or $R=8.66$ in the $r_{\mathrm{0}}$ scale. It
is interesting to compare the obtained $R$ with its value in the variational
analysis method used in [36]. Substituting $\eta=1$ and $s=0$ in the energy
functional in Eq. (20), we get

$\frac{E}{\epsilon}=\frac{3N_{\mathrm{0}}}{4R^{2}}+\frac{N_{\mathrm{0}}^{2}}{4\sqrt{2}\pi^{3/2}R^{3}}\\\
-\frac{N_{\mathrm{0}}^{2}}{8\pi
R}\int_{0}^{\infty}\operatorname{Erfc}\left(\frac{k_{*}}{\sqrt{2}}\right)dk_{*}.$

Its extremum is defined by the equation

$R^{2}-\frac{6\sqrt{2}\pi^{3/2}}{N_{\mathrm{0}}}R-3=0$ (23)

that gives $R=1.73$ or $R=7.86$ in the $r_{\mathrm{0}}$ scale. Thus,
$R_{\mathrm{c}}=0.9\,kpc$ (see Eq. (18)) and, therefore, the variational
analysis method and numerical calculation (see Fig. 1 (a)) are in a good
agreement.

####  III.2.2 Rotating axially-symmetric core

In the case $s=1$ (see Eq.(16)), we have a wave function, which depends on
cylindrical coordinates $r,z,\phi$

$\psi(r,\phi,z)=A\frac{r}{R}e^{-\frac{r^{2}}{2R^{2}}-\frac{z^{2}}{2(R\eta)^{2}}+i\phi}$
(24)

and the density function equals

$\rho(r,z)=\rho_{\mathrm{0}}\frac{r}{R}e^{-\frac{r^{2}}{R^{2}}-\frac{z^{2}}{(R\eta)^{2}}},$
(25)

where $A$ is given by Eq. (17).

Figure 1: Halo density profile $\rho/\rho_{0}$ as a function of dimensionless
$r/r_{0}$ coordinate in the plane $z=0$, both $x$ and $y$ axes have log scale.
The cyan insets in both plots show 3D density isosurfaces of the corresponding
BEC cores. Left panel (a) shows the halo with the BEC core in a soliton state
($s=0$). Three curves correspond to different values of parameter $\chi=4\pi
a_{\mathrm{s}}\hbar^{2}\rho_{\mathrm{c}}/(m^{2}k_{\mathrm{B}}T)$, so that
while increasing $\chi$ one decreases effective temperature $T$ and vice
versa. Right panel (b) shows the halo with the core in a vortex state ($s=1$,
$\chi$ = 20). Note, we investigate in detail the isothermal envelope for
$\chi$ = 20, which is consistent with observations for the Milky Way. The
black dashed line divides the distribution into two parts: the inner region
with a rotating core and the outer region composed of an isothermal envelope.

The dimensionless total energy in Eq. (20) for $s=1$ reads

$\frac{E}{\epsilon}=\frac{N_{\mathrm{0}}(1+4\eta^{2})}{4R^{2}\eta^{2}}+\frac{N_{\mathrm{0}}^{2}}{8\sqrt{2}\pi^{3/2}R^{3}\eta}\\\
-\frac{N_{\mathrm{0}}^{2}}{8\pi
R}\int_{0}^{\infty}\operatorname{Erfc}\left(\frac{k_{*}\eta}{\sqrt{2}}\right)\left(1-\frac{k_{*}^{2}}{4}\right)^{2}e^{-\frac{k_{*}^{2}(1-\eta^{2})}{2}}dk_{*}.$

Equations of an extremum of the total energy with respect to $\eta$ and $R$
yield the solution $\eta=1.464,$ and $R=1.226$ in the $L$ scale. In the
$r_{\mathrm{0}}$ scale, we have $R=5.57$. To determine the core density
distribution, we use the variational analysis result. We assume that the core
interacts negligibly weakly with the isothermal envelope. Therefore, for the
isothermal envelope region, we use the numerical distribution
$f_{\mathrm{N}}(r_{\mathrm{sph}})=f_{\mathrm{N}}(\sqrt{r^{2}+z^{2}})$ (see
Fig. 1 (a)), derived under $\mathbf{u}=0$ condition.

Thus, we obtain (see Fig. 1 (b))

$\rho(r,z)=\rho_{0}\begin{cases}1.92\frac{r}{R}e^{-\frac{r^{2}}{R^{2}}-\frac{z^{2}}{(R\eta)^{2}}},\frac{r_{\mathrm{sph}}}{r_{\mathrm{0}}}\leq\frac{R_{\mathrm{c}}}{r_{\mathrm{0}}}\\\
f_{\mathrm{N}}\left(\frac{r_{\mathrm{sph}}}{r_{\mathrm{0}}}\right),\frac{r_{\mathrm{sph}}}{r_{\mathrm{0}}}>\frac{R_{\mathrm{c}}}{r_{\mathrm{0}}},\end{cases}$
(26)

where $r=\sqrt{x^{2}+y^{2}}$ and $z$ are cylindrical coordinates and
$r_{\mathrm{sph}}=\sqrt{x^{2}+y^{2}+z^{2}}$. Here $\rho_{\mathrm{0}}$ is the
spherical halo central density. The spherically symmetric isothermal envelope
density
$\rho(r,z)=\rho_{0}f_{\mathrm{N}}\left({r_{\mathrm{sph}}}/{r_{\mathrm{0}}}\right)$
is found numerically by solving Eq. (11). The total core radius
$R_{\mathrm{c}}$ is defined by Eq. (18).

By using $\mathbf{u}=\frac{\mathbf{j}_{\mathrm{ph}}}{\rho}$ and the particle
current

$\mathbf{j}_{\mathrm{ph}}=-\frac{i\hbar}{2m}(\psi_{\mathrm{ph}}^{*}\mathbf{\nabla}\psi_{\mathrm{ph}}-\psi_{\mathrm{ph}}\mathbf{\nabla}\psi_{\mathrm{ph}}^{*})\\\
=\frac{\hbar}{m}\frac{|\psi_{\mathrm{ph}}|^{2}}{r}\mathbf{e}_{\phi},$

we find the velocity distribution $\mathbf{u}(\mathbf{r})$ of DM particles

$\mathbf{u}=\frac{\frac{\hbar}{m}\frac{|\psi_{\mathrm{ph}}|^{2}}{r}}{|\psi_{\mathrm{ph}}|^{2}}\mathbf{e}_{\phi}=\frac{\hbar}{m}\frac{1}{r}\mathbf{e}_{\phi}=\alpha\frac{cr_{\mathrm{0}}}{r}\mathbf{e}_{\phi},$
(27)

where $\alpha=\hbar/(mr_{\mathrm{0}}c)=0.31\cdot 10^{-3}$. Obviously, the
velocity of condensate particles increases while approaching the center of the
vortex. Note that there is an inner region where the velocity becomes of the
order of $c$ and, therefore, this region cannot be described by making use of
the gravitoelectromagnetism ansatz (see Appendix A for explanation). This
region is limited by the radial distance $r=\alpha r_{\mathrm{0}}=2.2\times
10^{-5}\,kpc$.

In the two following sections, by using the formalism of GEM, we describe
particle movement in the gravitational field of DM in the $s=0,1$ states
aiming to understand how baryonic matter particles interact with the proposed
DM .

##  IV Gravielectric field and rotation curves

In this section, we obtain numerical results for the gravielectric (Newtonian)
component of the DM halo gravitational field. Having calculated the field, we
analyze the rotation curves, predicted by the model in the cases of soliton
and vortex core.

To determine the gravielectric potential in the case of a non-rotating halo we
use the numerically obtained density distribution (see Fig. 1 (a)). In the
case of a rotating axially symmetric halo, the mass density distribution is
shown in Fig. 1 (b).

Figure 2: The radial component of gravielectric field
$a_{\mathrm{E}}/a_{\mathrm{0}}$ (blue dashed line) and density (red solid
line) of the non-rotating halo ($s=0$ core) as functions of the dimensionless
$r/r_{\mathrm{0}}$ coordinate, both $x$ and $y$ axes have log scale. Here
$a_{\mathrm{0}}=5.38\times 10^{-13}\frac{km}{s^{2}}$, $r_{0}=71pc$.

In the spherically symmetric case of non-rotating halo ($s=0$), only the
radial component of the gravielectric field is not zero (see Eq.(7)) and the
corresponding gravielectric acceleration
$\mathbf{a}_{\mathrm{E}}=-\mathbf{E}_{\mathrm{g}}=a_{\mathrm{E}}\mathbf{e}_{r}$
(see Eq.(9)) is presented in Fig. 2. The acceleration at large distances
behaves like $a_{\mathrm{E}}/a_{\mathrm{0}}=82.66r_{\mathrm{0}}/r$, i.e.,
$a_{\mathrm{E}}=9.3\times 10^{-29}\frac{\,kpc^{2}}{s^{2}}\times 1/r$. Here
$a_{\mathrm{0}}=G\rho_{\mathrm{0}}r_{\mathrm{0}}=5.38\times 10^{-13}km/s^{2}$.
In the core region, where the density distribution is described by the
variational ansatz (22), the gravielectric potential and the corresponding
acceleration can be found analytically

$\frac{1}{r^{2}}\frac{\partial}{\partial r}r^{2}\frac{\partial}{\partial
r}\Phi_{\mathrm{g}}=-4\pi G\rho_{\mathrm{0}}e^{-\frac{r^{2}}{R^{2}}}.$

The general solution is given by

$\Phi(a)=-4\pi
G\rho_{\mathrm{0}}R^{2}\left(\frac{c_{\mathrm{1}}}{r}+c_{\mathrm{2}}-\frac{R\sqrt{\pi}\operatorname{Erf}(r/R)}{4r}\right).$

where $\operatorname{Erf}(x)$ denotes the error function and $c_{1}$, $c_{2}$
are constants.

Figure 3: The radial component of gravielectric acceleration
$a_{\mathrm{Er}}/a_{\mathrm{0}}$ (blue dashed line) and density (red line) of
the rotating halo ($s=1$ core) as functions of dimensionless
$r/r_{\mathrm{0}}$ coordinate in the $z=0$ plane, both $x$ and $y$ axes have
log scale. Here $a_{\mathrm{0}}=5.38\times 10^{-13}\frac{km}{s^{2}}$,
$r_{0}=71pc$. Figure 4: The radial component of gravielectric acceleration
$a_{\mathrm{Er}}/a_{\mathrm{0}}$ induced by the rotating halo ($s=1$ core) as
a function of dimensionless $r/r_{\mathrm{0}}$ and $z/r_{\mathrm{0}}$
coordinates. Here $a_{\mathrm{0}}=5.38\times 10^{-13}\frac{km}{s^{2}}$,
$r_{0}=71pc$. The left panel shows the isothermal envelope region with the
three axes in the log scale and the right panel is a zoom-in of the core
region. Figure 5: The $z$-component of gravielectric acceleration
$a_{\mathrm{Ez}}/a_{\mathrm{0}}$ induced by the rotating halo ($s=1$ core) as
a function of dimensionless $r/r_{\mathrm{0}}$ and $z/r_{\mathrm{0}}$
coordinates. Here $a_{\mathrm{0}}=5.38\times 10^{-13}\frac{km}{s^{2}}$,
$r_{0}=71pc$. The left panel shows the isothermal envelope region with the
three axes in the log scale and the right panel is a zoom-in of the core
region.

We can set $c_{\mathrm{2}}=0$. At a large distance, the gravielectric
potential of the halo must be equal to the potential of a body with the same
mass $M=\pi^{3/2}\rho_{\mathrm{0}}R^{3}$. This implies that
$c_{\mathrm{1}}=0$. Thus, $\Phi(r)$ is completely determined and we have the
radial acceleration

$\mathbf{a}_{\mathrm{E}}(r)=\nabla\Phi_{\mathrm{g}}(r)\\\ =\pi
G\rho_{\mathrm{0}}R^{3}\left(\frac{2e^{-r^{2}/R^{2}}}{Rr}-\frac{\sqrt{\pi}\operatorname{Erf}\left({\frac{r}{R}}\right)}{r^{2}}\right)\mathbf{e}_{r}.$

Clearly, $a_{\mathrm{E}}$ has a maximum at $r=R=8.66$ in the $r_{\mathrm{0}}$
scale in agreement with the radial gravielectric acceleration shown in Fig. 2.

Gravielectric field in the case of vortex core has radial and $z$ components
in cylindrical coordinates, namely
$\mathbf{a}_{\mathrm{E}}=a_{\mathrm{Er}}\mathbf{e}_{r}+a_{\mathrm{Ez}}\mathbf{e}_{z}$.
They are illustrated in Figs. 4 and 5, respectively. The radial dependence of
the gravielectric radial acceleration in the $z=0$ plane is shown in Fig. 3.
Notice that at $r\approx 0.81r_{\mathrm{0}}=0.058\,kpc$ the acceleration
projection changes sign, hence test particles are repelled in the interior
region and attracted in the exterior region. This result stems from the
geometry of the considered doughnut-shaped halo with a hole.

Figure 6: The rotation (Kepler) velocity $v$ in the $z=0$ plane as a function
of the radial distance $r$. The pink dashed line corresponds to the non-
rotating spherical halo ($s=0$ core) and the cyan solid line to the rotating
halo ($s=1$ core). The background represents a gradient plot of the density
distribution in $s=1$ case

Now we aim to determine the impact of the gravitational field of the DM halo
on the movement of celestial bodies in the Milky Way galaxy. According to our
model (see Sec. III) density distribution depends on the state of the core,
which must lead to a difference between the rotation curves, which they
induce. To demonstrate how the gravielectric acceleration induces rotation in
the $s=0$ and $s=1$ cases, we present the rotation velocity $v$ in the $z=0$
plane as a function of the radial distance $r$ in Fig. 6. The new result here
is the curve in the case $s=1$, while $s=0$ case was discussed earlier in
[16]. The two halos with $s=0$ and $s=1$ core have equal mass, which is the
observed mass of DM halo in the Milky Way, according to the model discussed in
Sec. III. The numerical results indeed show that at large distances the
corresponding rotational curves have the same asymptotic. Note that, the
gravielectric force in $s=1$ case changes its sign at
$r=0.81r_{\mathrm{0}}=0.058\,kpc$. Hence, at distances less than $0.058\,kpc$
there are no stable rotation orbits in the rotating halo model. However, the
stable orbits are possible if one includes not only DM but also the other
sources of the gravitational field, namely, the baryonic galactic bulge and
the supermassive black hole in the central region of the galaxy.

##  V Gravimagnetic field in the BEC core

In this section, we obtain numerical results for the gravimagnetic (first
post-Newtonian) component of the DM halo gravitational field (see Subsec. II.2
of Sec.II). The component is induced by a moving source, hence, it is nonzero
only in the second case of the DM halo with a vortex core.

To determine the gravimagnetic potential in the case of a rotating axially
symmetric halo we use the mass density and velocity distributions given by
Eqs. (26) and (27). The calculation is based on Eqs. (8) and (6). The results
of numerical integration for radial and $z$-components of the gravimagnetic
field, $\mathbf{B}=B_{r}\mathbf{e}_{r}+B_{z}\mathbf{e}_{z}$, are shown in
Figs. 7 and 8, respectively. Fig. 9 displays the $z$-component of the
gravimagnetic field $B_{z}$ in the $z=0$ plane (the radial component of the
gravimagnetic field equals zero in this plane).

Figure 7: The radial component of gravimagnetic field $B_{r}/a_{\mathrm{0}}$
induced by the rotating core as a function of dimensionless $r/r_{\mathrm{0}}$
and $z/r_{\mathrm{0}}$ coordinates. Here $a_{\mathrm{0}}=5.38\times
10^{-13}\frac{km}{s^{2}}$, $r_{0}=71pc$. Figure 8: The $z$-component of
gravimagnetic field $B_{z}/a_{\mathrm{0}}$ induced by the rotating core as a
function of dimensionless $r/r_{\mathrm{0}}$ and $z/r_{\mathrm{0}}$
coordinates. Here $a_{\mathrm{0}}=5.38\times 10^{-13}\frac{km}{s^{2}}$,
$r_{0}=71pc$. Figure 9: The $z$-component of the gravimagnetic field
$B_{z}/a_{\mathrm{0}}$ (blue dashed line) and density (red line) of the
rotating core as functions of dimensionless $r/r_{\mathrm{0}}$ in $z=0$ plane.
Here $a_{\mathrm{0}}=5.38\times 10^{-13}\frac{km}{s^{2}}$, $r_{0}=71pc$.

Having determined the gravimagnetic field, we can calculate the corresponding
acceleration of the test particle. Using Eq.(9), we have

$\mathbf{a}_{\mathrm{B}}(\mathbf{r}=(a,b,k))=-\frac{2}{c}v\mathbf{e}_{\phi}\times\mathbf{B}_{\mathrm{g}}\\\
=-1.38\alpha
G\rho_{\mathrm{0}}r_{\mathrm{0}}\frac{v}{c}(B_{r}(a,b,k)\mathbf{e}_{r}+B_{z}(a,b,k)\mathbf{e}_{z})\\\
=a_{\mathrm{Br}}(a,b,k)\mathbf{e}_{r}+a_{\mathrm{Bz}}(a,b,k)\mathbf{e}_{z}.$

where $a=r/r_{0}$, $b=\phi$, $c=z/r_{0}$ are rescaled cylindrical coordinates.

This allows us to estimate the impact of the gravimagnetic field on stars’
motion. In the case of the Milky Way galaxy,
$v=v_{\mathrm{0}}+\gamma_{\mathrm{0}}r_{\mathrm{0}}a$ if
$a<a_{\mathrm{break}}$ and $v=v_{1}+\gamma_{1}r_{\mathrm{0}}a$ for $a\geq
a_{\mathrm{break}}$ [42]. Constants $\gamma_{0}$, $\gamma_{1}$,
$a_{\mathrm{break}}$, and $v_{1}$ are different for the thick and thin
galactic disks’ velocity profiles. Setting
$R_{\mathrm{break}}=r_{\mathrm{0}}a_{\mathrm{break}}=5\,kpc$ and
$v_{\mathrm{0}}$=0 in both cases gives the values of parameters presented in
Table I. This approximation is valid up to $13\,kpc=180r_{\mathrm{0}}$ [42].
We should emphasize here that $v$ includes only the component of velocity
directed along $\mathbf{e}_{\phi}$ and does not include the component along
$\mathbf{e}_{r}$. It is important to distinguish the $\phi$-component and the
absolute value of the whole velocity when dealing with sufficiently non-
circular elliptic orbits.

Table 1: Parameters of the Milky Way’s rotational velocity profiles [42]. Galactic disk |  $v_{1}$ $[kms^{-1}]$ | $\gamma_{0}[kms^{-1}kpc^{-1}]$ | $\gamma_{1}[kms^{-1}kpc^{-1}]$
---|---|---|---
thin disk | 236.71 | 45.41 | -1.93
thick disk | 206.93 | 39.086 | -2.30

According to Eq.(9), the gravimagnetic acceleration in galactic plane $c=0$
can be estimated as

$\mathbf{a}_{\mathrm{B}}=-1.38\alpha
G\rho_{0}r_{\mathrm{0}}\frac{v_{i}+\gamma_{i}r_{0}a}{c}B_{r}(a,b,0)\mathbf{e}_{r},$

where $i=0$ for $a<a_{\mathrm{break}}$ and $i=1$ for $a\geq
a_{\mathrm{break}}$. The corresponding plot is shown in Fig. 10. The spike on
the red curve, which shows the modulus of the ratio of the gravimagnetic
acceleration to the gravielectric one $|a_{\mathrm{Br}}/a_{\mathrm{Er}}|$
appears because the gravielectric acceleration changes sign at
$r=0.81r_{0}=0.058\,kpc$.

Figure 10: The radial component of gravimagnetic acceleration (solid and
dashed blue lines) $a_{\mathrm{Br}}/a_{\mathrm{0}}$ for thin and thick disks,
respectively, and the absolute value of the ratio of gravimagnetic
acceleration to gravielectric $|a_{\mathrm{Br}}/a_{\mathrm{Er}}|$ (both for
thin and for thick disks) as a function of $r/r_{\mathrm{0}}$. Here
$a_{\mathrm{0}}=5.38\times 10^{-13}\frac{km}{s^{2}}$, $r_{0}=71pc$. Figure 11:
The radial component of gravimagnetic acceleration (dashed blue line)
$a_{\mathrm{Br}}/a_{\mathrm{0}}$ and gravielectric acceleration (red line)
$a_{\mathrm{Er}}/a_{\mathrm{0}}$ in the inner region of halo. Here the grey
region corresponds to $r<0.1r_{\mathrm{0}}$, where the gravimagnetic
approximation is not valid. Here $a_{\mathrm{0}}=5.38\times
10^{-13}\frac{km}{s^{2}}$, $r_{0}=71pc$.

It is interesting that $a_{\mathrm{Br}}(a)$ tends to a constant in the $a\ll
1$ limit (see Fig. 11). This directly follows from the analytical expression.
In the interior region $r<5\,kpc$, we have

$\frac{a_{\mathrm{Br}}}{a_{\mathrm{0}}}=-1.38\alpha\frac{\gamma_{0}r_{\mathrm{0}}a}{c}B_{r}(a,b,0).$

In the $a\ll 1$ limit, we find

$B_{r}(a,b,0)\\\
\approx\frac{2\pi}{a}\int_{0}^{\infty}dx\int_{-\infty}^{\infty}dz\frac{x^{2}}{\sqrt{x^{2}+z^{2}}}e^{-\frac{x^{2}}{R^{2}}-\frac{z^{2}}{(R\eta)^{2}}}.$

The last integral can be calculated numerically which yields

$\frac{a_{\mathrm{Br}}}{a_{\mathrm{0}}}\approx-1.38\alpha\frac{\gamma_{0}r_{0}}{c}\times
247=9.8\times 10^{-7}.$

We see that in the case under consideration the gravimagnetic acceleration
indeed tends to be a constant in the $a\ll 1$ limit.

The gravimagnetic field calculations performed in this section allow us to
obtain some testable predictions of the model. According to numerical results
for $\mathbf{B}_{\mathrm{g}}$ and $\mathbf{E}_{\mathrm{g}}$, the gravielectric
force changes its sign at $r=0.81r_{0}=0.058\,kpc$, and the gravimagnetic
force component is attractive or repulsive, depending on the direction of the
motion. The acceleration in the polar coordinates $(r,\phi)$ is given by
$\mathbf{a}=(\ddot{r}-r\dot{\phi}^{2})\mathbf{e}_{r}+(r\ddot{\phi}+2\dot{r}\dot{\phi})\mathbf{e}_{\phi}$.
Then the equations of motion for a star take the form

$\frac{d^{2}r}{dt^{2}}=r\left(\frac{d\phi}{dt}\right)^{2}-E_{r}-\frac{2B_{z}r}{c}\frac{d\phi}{dt},$
(28)
$r\frac{d^{2}\phi}{dt^{2}}=\frac{2B_{z}}{c}\frac{dr}{dt}-2\frac{dr}{dt}\frac{d\phi}{dt}.$
(29)

Since the gravielectric acceleration dominates over the gravimagnetic one, it
suffices to take the latter into account as a perturbation. Therefore, we
treat $B_{\mathrm{g}}$ as the first-order perturbation and expand $\phi(t)$
and $r(t)$ around the solution $r_{\mathrm{c}}$ and $\phi_{\mathrm{c}}$
determined by the gravielectric acceleration. For $r=r_{\mathrm{c}}+\delta r$
and $\phi=\phi_{\mathrm{c}}+\delta\phi$, in the zeroth-order, we have the
Kepler problem equations with $E_{r}(r_{\mathrm{c}})$ calculated numerically
in Sec.IV. The corresponding solutions are elliptic orbits. For simplicity, we
will consider only the case of circular orbits
$r_{\mathrm{c}}(\phi)=r_{\mathrm{c}}=const$. By substituting
$E_{r}(r_{\mathrm{c}}+\delta r)\approx
E(r_{\mathrm{c}})+\frac{dE}{dr}(r_{\mathrm{c}})\delta r$ in Eqs. (28) and
(29), we obtain

$\frac{d^{2}\delta r}{dt^{2}}=w_{0}^{2}\delta
r+2r_{\mathrm{c}}w_{0}\frac{d\delta\phi}{dt}-\frac{dE_{r}}{dr}\Big{|}_{r_{\mathrm{c}}}\delta
r-\frac{2B_{z}}{c}r_{\mathrm{c}}w_{0},$
$r_{\mathrm{c}}\frac{d^{2}\delta\phi}{dt^{2}}=-2w_{0}\frac{d\delta r}{dt}.$

where $w_{0}=\frac{d\phi_{\mathrm{c}}}{dt}$ is angular frequency, induced by
gravielectric field. Thus, it can be explicitly written as
$w_{0}^{2}=\frac{E_{r}(r_{\mathrm{c}})}{r_{\mathrm{c}}}$ Integrating the
second equation, we get

$\frac{d\delta\phi}{dt}=-\frac{2w_{0}}{r_{\mathrm{c}}}\delta r,$

where we set the integration constant to zero. Substituting this relation in
the first equation, we find

$\frac{d^{2}\delta
r}{dt^{2}}=-\left(3w_{0}^{2}+\frac{dE_{r}}{dr}\Big{|}_{r_{\mathrm{c}}}\right)\delta
r-\frac{2B_{z}}{c}r_{\mathrm{c}}w_{0}.$

From the numerical result, we see that $f(E)$ is positive and tends to zero at
a large distance. Then, for
$3\frac{E_{r}(r_{\mathrm{c}})}{r_{\mathrm{c}}}+\frac{dE_{r}}{dr}\Big{|}_{r_{\mathrm{c}}}=\Omega^{2}>0$,
we find solutions

$\delta
r=-\frac{2B_{z}r_{\mathrm{c}}w_{0}}{c\Omega^{2}}+J\sin(\Omega(t-t_{0})),$
$\delta\phi=\delta\phi_{\mathrm{c}}+\frac{4B_{z}w_{0}^{2}}{c\Omega^{2}}t+\frac{2w_{0}J}{\Omega
r_{\mathrm{c}}}\cos(\Omega(t-t_{0})),$

where $\delta r_{\mathrm{c}}$, $J$, and $t_{0}$ are defined by the
corresponding initial conditions.

It is interesting to estimate $\Omega^{2}$ at some distance $r_{\mathrm{c}}$,
e.g., $r_{\mathrm{c}}=8\,kpc=113r_{\mathrm{0}}$ which is the distance of the
Sun from the center of the galaxy. Then we have $\Omega=\sqrt{0.0017\times
a_{0}/r_{0}}=6.48\times 10^{-16}s^{-1}$ (the corresponding period is
$T=3.1\times 10^{8}\,y$), $B_{z}=2.76\times 10^{-8}a_{0}$, and $\delta
r=-2B_{z}r_{\mathrm{c}}w_{0}/c\Omega^{2}=-2.7\times 10^{-8}r_{0}=-0.38a.u.$.
The latter distance is approximately equal to 80 solar radii. The angular
frequency is shifted by the value $4B_{z}w_{0}^{2}/c\Omega^{2}=4.8\times
10^{-25}s^{-1}$ (the corresponding period is $T=4.2\times 10^{17}y$).

##  VI Conclusions

We investigated the model of DM halo with BEC core composed of ultra-light
bosonic particles. Solving the generalized GPP equations for self-gravitating
BEC we obtained the density profile of the DM halo and analyzed its core and
envelope structure. The density and velocity profiles were found for two types
of stable structures with topological charges ($s=0$ and $s=1$) of the BEC
core.

Using this DM halo description, we investigated its gravitational field and
the impact of this field on the baryonic matter. The key result of our paper
is that the observable effects, predicted by the ULDM halo model, depend on
the state of the core. In particular, solitonic and vortex cores yield
different density and velocity distributions and thus different gravitational
fields. The doughnut-like density distribution (vanishing at the vortex core)
and vortex flows (rapidly increasing at the vortex axis) of the BEC core can
significantly modify both gravielectric and gravimagnetic components of the
gravitational field. We described the gravitational fields of these two core
configurations by using the gravimagnetism approach. A dominant component of
the gravitational field is the gravielectric (Newtonian) one, which generates
the rotation of celestial bodies in the galaxy. The rotational velocity
induced by the halo with vortex is smaller close to the core region but has
the same asymptotics at large distances in comparison with the non-rotating
halo.

The first post-Newtonian component of the gravitational field, which is called
gravimagnetic, is induced by the rotation of the BEC vortex core and appears
only in the model of a rotating halo. Although, as expected, the gravimagnetic
acceleration is much weaker than the gravielectric one, it can affect the
dynamics of baryonic matter in the halo, especially in its inner region. In
our simplified perturbation approach for circular orbit gravimagnetic field
yields radius and frequency shift, and can also induce trajectory
oscillations, depending on initial conditions.

There are several possible directions in which the present study could be
extended. An analysis of gravitational fields beyond the gravimagnetic
approach is required in the central region of the galaxy, due to the high
rotational velocity of BEC there. Furthermore, according to astrophysical
observations, there is a supermassive black hole in the center of our galaxy
whose presence should be taken into account. Finally, the gravitational
effects of baryonic matter should be included in further studies.

##  VII Acknowledgments

The authors are grateful to Yelyzaveta Nikolaieva, Sebastian Ulbricht,
Stanislav Vilchinskii, and Luca Salasnich for useful discussions and comments.
A.Y. acknowledge support from BIRD Project ”Ultracold atoms in curved
geometries” of the University of Padova.

## Appendix A

Let us discuss the self-consistency of our model, which makes use of the GEM
approach to describe the first post-Newtonian contribution to the
gravitational field potential. We assumed that a test particle (celestial body
acted upon by the gravitational field) propagates with a non-relativistic
speed $v$ so that all terms of higher than linear order in $O(v/c)$ can be
neglected in the equations of motion. As to DM, we describe it by using the
nonlinear Schrödinger equation with gravitational potential
$\Phi_{\mathrm{g}}$.

Since the hydrodynamical velocity in the vortex (the state with $s=1$) is
$u(r)=\alpha{cr_{0}}/{r}$, it increases at small $r$ and attains at
$r\sim\alpha r_{0}$ values of the order of $c$. Obviously, the Newtonian
treatment is not applicable in this region. Therefore, we use the Klein-Gordon
equation in order to describe the relativistic equation of motion of bosons,
as follows:

$\nabla_{\alpha}\nabla^{\alpha}\phi+\left[\left(\frac{mc}{\hbar}\right)^{2}-U(|\phi|^{2})\right]\phi=0,$
(30)

where $U=\frac{2m}{\hbar^{2}}gN|\phi|^{2}$ and $\phi$ is the scalar field. We
neglect the effective temperature because only the core region is investigated
(the hydrodynamical velocity $u(r)$ is nonzero only in the core region) and
$\nabla_{\alpha}$ denotes covariant derivative in curved space-time.

The metric in the GEM approach reads (here all notations are the same as in
Subsec. II.2)

$dS^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=\left(1-\frac{2\Phi_{g}}{c^{2}}\right)(dx^{0})^{2}\\\
+\frac{4}{c^{2}}\left(\mathbf{A}_{g}\mathbf{dx}\right)dx^{0}+\left(-1-\frac{2\Phi_{g}}{c^{2}}\right)\delta_{ij}dx^{i}dx^{j}$

and the Laplace operator is given by

$\nabla_{\alpha}\nabla^{\alpha}\phi=\frac{1}{\sqrt{-g}}\partial_{\alpha}(\sqrt{-g}g^{\alpha\beta}\partial_{\beta}\phi)$

where $g=det(g_{\mu\nu})\approx-1$. Then we have

$\nabla_{\alpha}\nabla^{\alpha}\phi=\frac{1}{c^{2}}\left(1-\frac{2\Phi_{g}}{c^{2}}\right)\partial^{2}_{t}\phi-\frac{2A^{i}_{g}}{c^{3}}\partial_{t}\partial_{i}\phi\\\
-\frac{2}{c^{3}}\partial_{i}(A^{i}_{g}\partial_{t}\phi)-\partial_{i}\left[\left(1+\frac{2\Phi_{g}}{c^{2}}\right)\delta^{ij}\partial_{j}\phi\right],$

where fields $\Phi_{g}$ and $\mathbf{A}_{g}$ are time-independent. Taking into
account the gauge condition $\partial_{i}A^{i}_{g}=0$, we find

$\nabla_{\alpha}\nabla^{\alpha}\phi=\frac{1}{c^{2}}\left(1-\frac{2\Phi_{g}}{c^{2}}\right)\partial^{2}_{t}\phi\\\
-\frac{4A^{i}_{g}}{c^{3}}\partial_{t}\partial^{i}\phi+\partial_{i}\left[\left(1+\frac{2\Phi_{g}}{c^{2}}\right)\partial_{i}\phi\right].$

To obtain a nonrelativistic approximation of the Klein-Gordon equation we
represent the scalar field in the form $\phi=e^{imc^{2}t/\hbar}\psi$.
Substituting this expression in the Klein-Gordon equation and multiplying by
$e^{-imc^{2}t/\hbar}$ we get

$\frac{1}{c^{2}}\left(1-\frac{2\Phi_{g}}{c^{2}}\right)\left[\partial_{t}^{2}\psi+\frac{2imc^{2}}{\hbar}\partial_{t}\psi-\left(\frac{mc^{2}}{\hbar}\right)^{2}\psi\right]\\\
-\frac{4A^{i}_{g}}{c^{3}}\left[\partial_{t}\partial^{i}\psi+\frac{imc^{2}}{\hbar}\partial^{j}\psi\right]+\left[1+\frac{2\Phi_{g}}{c^{2}}\right]\partial^{j}\partial_{j}\psi\\\
+\frac{2}{c^{2}}\partial^{j}\Phi_{g}\partial_{j}\psi+\left[\left(\frac{mc}{\hbar}\right)^{2}-\frac{2m}{\hbar^{2}}U(|\psi|^{2})\right]\psi=0.$

Neglecting terms of order of $(u/c)^{2}$ and higher ($A_{g}\sim u/c$), we
obtain

$\frac{2im}{\hbar}\partial_{t}\psi-\left(\frac{mc}{\hbar}\right)^{2}\psi+2\Phi_{g}\left(\frac{m}{\hbar}\right)^{2}\psi+\partial^{j}\partial_{j}\psi\\\
+\left[\left(\frac{mc}{\hbar}\right)^{2}-\frac{2m}{\hbar^{2}}U(|\psi|^{2})\right]\psi=0$

Finally, after some straightforward simplifications, we derive the Schrödinger
equation in the form

$i\hbar\partial_{t}\psi=\left(-\frac{\hbar^{2}}{2m}\partial^{j}\partial_{j}+m\Phi_{g}+U(|\psi|^{2})\right)\psi.$

Thus, we conclude that the model is self-consistent if we take into account
only terms up to $u/c$, or, equivalently, in the region, where the
hydrodynamical velocity of vortex is not relativistic ($u\ll c$).

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# Enabling Quantum Natural Language Processing for Hindi Language

Naman Srivastava
IIIT Dharwad
Dharwad, India
<EMAIL_ADDRESS>
&Gaurang Belekar
IIIT Dharwad
Dharwad, India
<EMAIL_ADDRESS>
&Sunil Saumya
IIIT Dharwad
Dharwad, India
<EMAIL_ADDRESS>
Aswath Babu H.
IIIT Dharwad
Dharwad, India
<EMAIL_ADDRESS>

###### Abstract

Quantum Natural Language Processing (QNLP) is taking huge leaps in solving the
shortcomings of classical Natural Language Processing (NLP) techniques and
moving towards a more "Explainable" NLP system. The current literature around
QNLP focuses primarily on implementing QNLP techniques in sentences in the
English language. In this paper, we propose to enable the QNLP approach to
HINDI, which is the third most spoken language in South Asia. We present the
process of building the parameterized quantum circuits required to undertake
QNLP on Hindi sentences. We use the pregroup representation of Hindi and the
DisCoCat framework to draw sentence diagrams. Later, we translate these
diagrams to Parameterised Quantum Circuits based on Instantaneous Quantum
Polynomial (IQP) style ansatz. Using these parameterized quantum circuits
allows one to train grammar and topic-aware sentence classifiers for the Hindi
Language.

Enabling Quantum Natural Language Processing for Hindi Language

Naman Srivastava IIIT Dharwad Dharwad, India<EMAIL_ADDRESS>Gaurang
Belekar IIIT Dharwad Dharwad, India<EMAIL_ADDRESS>Sunil Saumya IIIT
Dharwad Dharwad, India<EMAIL_ADDRESS>

Aswath Babu H. IIIT Dharwad Dharwad, India<EMAIL_ADDRESS>

##  1 Introduction

In the ever-evolving landscape of information technology and computational
linguistics, Quantum Natural Language Processing (QNLP) represents a
groundbreaking paradigm shift at the intersection of Quantum Computing and
natural language understanding. As the world delves deeper into the realms of
Quantum Mechanics, harnessing the power of Quantum Computing to enhance the
processing and comprehension of human language has emerged as a promising
frontier. QNLP is a burgeoning field that promises to revolutionize how we
interact with and extract knowledge from textual data, opening up new horizons
for applications ranging from machine translation and sentiment analysis to
information retrieval and cognitive computing. The first high-level Python
library for QNLP was presented in 2021 as an open-source toolkit (Kartsaklis
et al., 2021).

Traditional Natural Language Processing (NLP) has made remarkable strides in
automating tasks like text classification, machine translation, and sentiment
analysis. However, it still grapples with inherent limitations, such as the
inability to process and comprehend the vast nuances of human language
efficiently. On the other hand, the model that interprets the Quantum
Mechanical Phenomena is found to be equivalent to the model of natural
language, which entails QNLP to be a quantum native of language processing.
The processing of linguistic structure can be easily encoded through a quantum
regime, whereas encoding grammar in classical means is very tedious and
costly.

Quantum Computing, on the other hand, leverages the principles of
superposition and entanglement, enabling it to handle complex computational
tasks exponentially faster than classical computers. Performing Natural
Language Processing (NLP) on Hindi text is significant due to the language’s
vast user base, cultural relevance, and economic opportunities. It can offer
insights into sentiments, opinions, and societal nuances, aids in governance
and policy analysis, supports content creation and localization, enhances
education, and has applications in healthcare, media analysis, linguistic
research, and disaster response.

In particular, the QNLP pipeline relies on diagrammatic representation,
wherein it involves first parsing an input sentence followed by generating its
associated string diagram. This diagram serves as the basis for constructing a
trainable Quantum Circuit or a Parameterized Quantum Circuit (PQC). The same
method is implemented here for the case of Hindi sentences. This research
endeavor delves into the methodology for crafting a Parameterized Quantum
Circuit from Hindi sentences, establishing a fundamental framework for the
development and training of a Quantum Natural Language Model tailored to the
Hindi language. This work is inspired by the recent work on pregroup
representation of Language (Debnath and Shrivastava, 2019; Abbaszade et al.,
2021). We use the pregroup grammar formalism discussed by (Debnath and
Shrivastava, 2019) to define the word interactions and assign the pregroup
notations.

##  2 Background

###  2.1 Quantum Computing

Quantum Computing is a field of computing technology that uses the principles
of Quantum Mechanics to perform certain types of calculations much faster than
traditional computers. In a regular computer, information is processed using
bits, which can be either 0 or 1. Quantum Computers, on the other hand, use
quantum bits or "Qubits" denoting the combination of both 0 and 1. What makes
Qubits special is that they can exist in multiple states at once. This ability
to exist in multiple states at once, known as superposition, allows Quantum
Computers to consider many possible solutions to a problem simultaneously.
Additionally, qubits can be "entangled," meaning the state of one qubit can
instantly affect the state of another, irrespective of their physical
separation, despite not having any sort of communication channel. This
property of entanglement enables Quantum Computers to solve certain complex
problems much faster than Classical Computers.

####  2.1.1 Dirac Notation

Dirac Notation, also known as bra-ket notation, is a mathematical notation
commonly used in Quantum Mechancs and Quantum Computing to represent and
manipulate quantum states and operations. It was developed by physicist Paul
Dirac and provides a concise and powerful way to describe quantum systems and
their transformations:

1\. A quantum state, such as the state of a qubit, is represented by a "ket"
vector, written as $|\psi\rangle$. For example, $|0\rangle$ and $|1\rangle$
represent the two possible states of a single qubit, where $|0\rangle$
corresponds to the binary state 0, and $|1\rangle$ corresponds to the binary
state 1.

2\. The adjoint (complex conjugate plus transpose) of a ket vector is
represented by a "bra" vector, written as $\langle\psi|$. If the ket vector is
represented by a column vector then the bra would be a row vector ensuring the
possible matrix multiplication that is analog to the scalar product existing
among vectors.

3\. The inner product of two quantum states is denoted as
$\langle\alpha|\beta\rangle$, where $|\alpha\rangle$ and $|\beta\rangle$ are
ket vectors. This represents the probability that states $|\alpha\rangle$ and
$|\beta\rangle$ alike.

4\. Quantum operators, such as quantum gates, are represented by matrices and
are often applied to quantum states using the ket-bra notation. For example,
if you have a gate represented by the matrix U and you want to apply it to a
qubit in state $|\psi\rangle$, you can write it as $U|\psi\rangle$.

5\. Quantum measurements can be represented by projection operators, which are
expressed using the outer product of ket and bra. For example, if you want to
represent a measurement that projects a quantum state onto $|0\rangle$, and
the involved projection operator is $|0\rangle\langle 0|$.

Dirac Notation is especially useful in Quantum Computing because it provides a
clear and compact way to represent and work with quantum states and
operations. It simplifies complex calculations and transformations and makes
it easier to express quantum algorithms and quantum circuits.

####  2.1.2 Quantum Gates

Quantum gates are fundamental building blocks in Quantum Computing, just like
regular logic gates are in classical computing. They are used to manipulate
and process qubits in a Quantum Computer. Quantum gates as tools or
instructions that allow to change of the state of qubits. These gates perform
specific operations on qubits, such as flipping their state, rotating their
orientation, or creating entanglement between them. Each quantum gate has a
particular purpose, just like different tools in a toolbox are used for
different tasks.

For example, you have quantum gates called the X-gate, which is like a switch
that flips the state of a qubit from 0 to 1 or vice versa. There’s also the
Hadamard gate, which puts a qubit into a superposition of 0 and 1, allowing it
to be in both states simultaneously. Other gates, like the CNOT gate, acting
together with Hadamard create entanglement between two qubits, so that when
one changes, the other does too, no matter how far apart they are.

###  2.2 Quantum Natural Language Processing

In Quantum Computing, the information present in qubits is represented in the
form of vectors. A similar approach of using vectors (or vector spaces) has
also been used frequently in Natural Language Processing involving Word
Embeddings such as Word2Vec, and FastText. In recent years, the intersection
of Natural Language Processing (NLP) and Quantum Computing has seen notable
success, giving rise to a field known as QNLP. This hybrid domain harnesses
the potential of Quantum Mechanics to address crucial aspects of language
processing, encompassing various NLP tasks. The existing approaches within
QNLP range from purely theoretical demonstrations of quantum advantage to
practical implementations of algorithms on quantum hardware. It has many
applications, such as machine translation, text summarization, and chatbot
creation. QNLP aims to use Quantum Computers to solve NLP problems. The goal
of QNLP is to develop quantum algorithms that can outperform classical
algorithms for NLP tasks. To create an explainable NLP model, we must include
the grammatical and syntax of the language in consideration(Lambek, 1958;
Chomsky, 2002). To incorporate such properties of language needs the
Distributional Compositional Categorical (DisCoCat) model (Coecke et al.,
2010; Grefenstette and Sadrzadeh, 2011; Sadrzadeh et al., 2013). Recent works
have shown the implementation of QNLP techniques on sentence classification
tasks using the DisCoCat model (Meichanetzidis et al., 2023; Metawei et al.,
2023)

###  2.3 Pre Group Grammar

Pregroup grammar, or pregroup formalism, is a mathematical approach for
analyzing natural language syntax. It is developed by linguist Joachim Lambek,
it offers a more precise and elegant description of linguistic structure than
traditional context-free grammars (Lambek, 1958, 1999) . In this framework,
words are linked to abstract algebraic "types" representing grammatical
information, such as nouns and verbs. Rules in pregroup grammar are defined as
algebraic operations on these types, enabling structured composition of
linguistic expressions. This approach efficiently captures linguistic
phenomena, like word order and ambiguity, making it valuable in computational
linguistics and linguistic formalism. Pregroup grammar focuses on the
algebraic structure within language expressions, attracting research interest
in linguistics and computer science. Several languages such as Sanskrit,
Arabic, German, French and Japanese have adopted pregroup calculus (Cardinal,
2002; Sadrzadeh, 2010):

$a^{l}\cdot a\rightarrow 1\rightarrow a\cdot a^{r}$

where $a^{l}$ and $a^{r}$ denote the left and right adjoint of the type of
word "a" respectively. For example in the sentence "Ram likes School", subject
and object nouns are represented by "n" and the verb "likes" expects a right
and left adjoints as these nouns. Algebraic calculation of sentence would lead
to flow out of sentence denoted by "s" with $n^{l}\cdot n\rightarrow
1\rightarrow n\cdot n^{r}$.

Ram likes School  $n\mathstrut$
$n^{r}\mathstrut$$s\mathstrut$$n^{l}\mathstrut$ $n\mathstrut$ (1)

A detailed exploration of the grammatical structures and sentence compositions
of English and Hindi reveals a combination of similarities and disparities.
Both languages encompass fundamental building blocks such as nouns, verbs, and
adjectives, which serve as the foundational elements for constructing
sentences. These shared components facilitate basic communication and
comprehension across the two languages. However, where English and Hindi
diverge is in the inclusion of specific linguistic features that are unique to
each language. For instance, Hindi employs a concept known as "Karak," which
plays a crucial role in assigning roles and relationships to different
sentence elements. This is a distinct feature absent in English sentence
structure. Additionally, tense markers in Hindi such as "Hai", "Thaa" etc
convey temporal information differently than their English counterparts,
introducing another layer of complexity in the grammatical structure. To
facilitate a better understanding of these differences, Table 1 is presented,
illustrating the various pregroup notations used in both languages and their
corresponding sentence components.

Symbol | Meaning
---|---
$\pi$ | Personal Pronoun
n | Noun Phrase
p | Simple Predicate
a | Adjective
o | Object or Transitive Verb
$k_{1}$ | Karaka
$\rho$ | Sambandh
$\alpha$ | Optional Auxiliary
$\tau$ | Tense Marker
Table 1: Pregroup Notations

###  2.4 Sentence Diagrams

Sentence diagrams in QNLP are graphical representations that use the
principles of Quantum Mechanics to depict the syntactic and semantic structure
of sentences in natural language. Unlike traditional sentence diagrams, which
typically rely on tree structures to represent grammatical relationships, QNLP
sentence diagrams employ quantum-inspired principles to model the complexity
and entanglement of linguistic elements within a sentence. In QNLP, sentence
diagrams, words, or linguistic units are represented as quantum-like states
(see Figure 1) or vectors in a high-dimensional space, that is like
$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$. These states can exhibit
superposition, entanglement, and other quantum properties, allowing for a more
nuanced representation of the relationships between words and their syntactic
and semantic roles. The use of quantum principles in sentence diagrams can
enable more accurate modeling of phenomena like word ambiguity, context-
dependent meaning, and the interaction between words within a sentence. QNLP
sentence diagrams are part of the broader effort to apply Quantum Computing
and quantum-inspired techniques to natural language understanding, with the
goal of enhancing the processing and comprehension of human language.

Figure 1: Bra and Ket notations in Sentence Diagram: The quantum state
($|\psi\rangle$) and its effect (operator obtained as $\langle\psi|$ by taking
adjoint) is simply 180 degree rotation about the normal axis.

###  2.5 Quantum Circuits From Sentence Diagrams

To train our NLP model using sentence diagrams and harness the quantum
advantage, it’s essential to transform these diagrams into trainable
parameterised quantum circuits. A specified sequence of gates applied to
specific wires in a quantum circuit is known as "Ansatz". To convert a diagram
to a circuit, we have several methods or ansatz, including the Instantaneous
Quantum Polynomial (IQP) ansatz and Matrix Product States (MPS) approaches.
Additionally, there are hybrid Quantum Algorithms introduced earlier (Sim et
al., 2019), offering further options for this transformation. These various
ansatzes provide us with a range of tools to adapt sentence diagrams into
quantum circuits, ultimately enabling us to explore the quantum advantages in
natural language processing tasks (Tangpanitanon et al., 2022; Meichanetzidis
et al., 2020; Abbaszadeh et al., 2021; Meichanetzidis et al., 2023; Kartsaklis
et al., 2021).

Figure 2: Standard Quantum Natural Language Pipeline

##  3 Methodology

A standard QNLP Pipeline involves the following steps as described in Figure
2:

  * •

Input Sentence to String Diagram: Using pregroup grammar of a language we
assign the atomic type to the words and phrases. Then, we use the DisCoCat
framework to create string diagrams corresponding to a sentence as shown in
Figure 2, wherein the words are connected by cup-shaped wires called "cups"
realized using an entangled pair.

  * •

Optimising String Diagram: Once the string diagram for a sentence is created,
we optimize this diagram by reducing the number of cups so that the
dimensionality of the words becomes smaller, here word "jata" needs such
correction. This step assists in removing the unnecessary complexities in a
sentence diagram.

  * •

String Diagram to Parametersied Quantum Circuit: The transition of String
Diagrams to Parameterised Quantum Circuit (PQC) carried via standard ansatz
such as Instantaneous Quantum Polynomial Ansatz (IQP) or Matrix Products
States (MPS) etc.

  * •

Building and Training Model: Once the Parameterised Quantum Circuits for each
sentence have been built, we can train our prediction model in a similar
manner to how we train a classical NLP. Each circuit is assigned initial
parameters and then measured. This measurement output is processed further
according to the task at hand, such as comparing with the expected output as
part of supervised learning. We use a classical optimizer to modify the
parameters of the quantum circuits, thus training the model.

###  3.1 The Sentences

We take two simple sentences from the Hindi language to demonstrate how we can
build String Diagrams and Parameterised Quantum Circuits from Hindi sentences.
These two sentences are: \- "Main School Jata hu" which translates to "I go to
school".
\- "Mukesh ne Khana khaya" which translates to "Mukesh ate the food".

Main School jaata hu  $\pi\mathstrut$ $o\mathstrut$
$o^{r}\mathstrut$$\pi^{r}\mathstrut$$s\mathstrut$$\tau^{l}\mathstrut$
$\tau\mathstrut$ (2)

Mukesh ne khaana khaya  $n\mathstrut$ $k_{1}^{l}\mathstrut$
$k_{1}\mathstrut$$o\mathstrut$ $o^{r}\mathstrut$$n^{r}\mathstrut$$s\mathstrut$
(3)

###  3.2 Creating Sentence Diagram

Using the pregroup notations and word connection formulated using the pregroup
grammar for Hindi, we create the sentence diagrams for the sentences mentioned
in Sec 3.1. For the Sentence 2 we obtain Figure 3. Similarly for Sentence 3 we
obtain the Figure 4.

Figure 3: Diagram for Sentence 1, single wired states are of type b and
multiple wired are of type d. Figure 4: Diagram for Sentence 2, single wired
states are of type b and multiple wired are of type d.

###  3.3 Building the Parameterised Quantum Circuit

Quantum Theory, established as a process theory, employs a diagrammatic
language denoted by string diagrams. Specifically, in the experiments we
consider, we utilize pure quantum theory. Within this context, processes
manifest as unitary operations or quantum gates associated with circuits. The
monoidal structure, which facilitates parallel processes, is represented by
the tensor product. Sequential composition mirrors the sequence of quantum
gates.

Figure 5: Ansatz for word with single wire Figure 6: Ansatz for word with
multiple wire Figure 7: Ansatz for "Cups" or "Bell State"

Wires assigned the basic pregroup type $b$ are allocated $q_{b}$ qubits. Word-
states with a singular output wire obtained from $\langle 0|$, see Figure 5.
To prepare these states, we opt for a series of Unitary gates representing an
Euler decomposition. This entails a series of one-qubit unitaries
$R_{z}(\theta_{1})$ to $R_{z}(\theta_{3})$, which are a simple rotational
matrices capable of rotating the subjected quantum state vectors. Word-states
with multiple output wires evolve from multi-qubit states on $k$ qubits where
$k>1$, which are structured by an IQP-style circuit, see FIgure 6. The word-
circuit is divided into $d$ layers. Each layer starts with Hadamard gates and
then connects every adjacent pair of qubit wires with a controlled rotational
$CR_{z}(\theta)$ gate. As all $CR_{z}$ gates are commutative, it’s feasible to
view them as one layer. The Kronecker tensor for $n$\- single output wires of
type $b$ are connected to word states via a GHZ state. In essence, the GHZ
circuit produces the state
$\sum_{a=0}^{2^{b}}\otimes^{n}_{i=1}|bin(a)\rangle$, where "bin" is a binary
representation. For example $n=2$ correspond to Bell (Entangled) State
$|00\rangle+|11\rangle$.The pregroup type $b$ is mapped to $q_{b}$-many nested
Bell effects ($\langle 00|+\langle 11|$), executed as a CNOT, followed by a
Hadamard gate on the control qubit and post-selection on $|00\rangle$ as shown
in Figure 7.

Figure 8: PQC for "Main School Jata hu", end gates are measurements leading to
information to classical register "c" shown at the bottom. Figure 9: PQC for
"Mukesh ne khana khaya", end gates are measurements leading to information to
classical register "c" shown at the bottom.

##  4 Discussion and Future work

In this study, we have delved into the process of constructing trainable
parameterized quantum circuits from Hindi sentences, utilizing pre-group
grammar and the DisCoCat model. This methodology holds significant relevance
within the broader framework of a versatile QNLP Pipeline. Once we
successfully create parameterized quantum circuits for a curated dataset
comprising Hindi sentences, the next crucial step involves training these
circuits to develop a proficient QNLP Model. Looking forward, our research
aspirations encompass an extension of this work to devise a nuanced sentence
classifier for Hindi sentences, integrating topic and context awareness
through a QNLP approach. We draw inspiration from a study by (Metawei et al.,
2023), aiming to align our approach with their advancements in this domain.
However, we acknowledge the inherent complexities lying ahead in this
endeavour. Hindi exhibits a notably freer word order compared to English,
presenting a challenge in accommodating potential restrictions on word order
movements within the construction of the PQC. Addressing and overcoming these
intricacies are essential components of our ongoing and future research,
paving the way for a more refined and accurate QNLP approach tailored to the
nuances of the Hindi language.

##  5 Conclusion

In conclusion, this paper presents a significant advancement in the field of
QNLP by extending its applications to the Hindi language. By leveraging the
pregroup representation of Hindi and the DisCoCat framework, we have
successfully designed sentence diagrams, which are then translated into
Parameterised Quantum Circuits using the IQP style ansatz. The resulting
parameterised circuits enable the development of grammar-aware and topic-aware
sentence classifiers tailored to the nuances of the Hindi language. This novel
approach not only contributes to the growing body of QNLP research but also
addresses the need for more explainable NLP systems in languages beyond
English, which promises exciting possibilities for natural language
understanding and processing across diverse linguistic landscapes.

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