# Non-Markovian quantum interconnect formed by a surface plasmon polariton
waveguide

Chun-Jie Yang* — School of Physics, Henan Normal University, Xinxiang 453007,
China  Xin-Yue Liu  School of Physics, Henan Normal University, Xinxiang
453007, China  Si-Yuan Bai  Key Laboratory of Quantum Theory and Applications
of MoE, Lanzhou University, Lanzhou 730000, China  Lanzhou Center for
Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province,
Lanzhou University, Lanzhou 730000, China  Jun-Hong An* —<EMAIL_ADDRESS>Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University,
Lanzhou 730000, China  Lanzhou Center for Theoretical Physics, Key Laboratory
of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000,
China

###### Abstract

Allowing the generation of effective interactions between distant quantum
emitters (QEs) via flying photons, quantum interconnect (QI) is essentially a
light-matter interface and acts as a building block in quantum technologies. A
surface plasmon polariton (SPP) supported by a metallic waveguide provides an
ideal interface to explore strong light-matter couplings and to realize QI.
However, the loss of SPP in metal makes the mediated entanglement of the QEs
damp with the increase of the distance and time, which hinders its
applications. We propose a scheme of non-Markovian QI formed by the SPP of a
metallic nanowire. A mechanism to make the generated entanglement of the QEs
persistent is discovered. We find that, as long as bound states are formed in
the energy spectrum of total QE-SPP system, the damping of the SPP-mediated
entanglement is overcome even in the presence of the metal absorption to the
SPP. Our finding enriches our understanding of light-matter couplings in
absorptive medium and paves the way for using the SPP in designing QI.

##  I Introduction

Pursuing the utilization of quantum resources to make scientific and
technological innovations, a quantum revolution is in the making [1, 2]. With
the dramatic progress in individual quantum systems, including trapped ions,
neutral atoms, superconducting qubits, and photonics [3, 4, 5, 6], for quantum
computation and metrology, an urgent requirement is to integrate them to form
advanced technologies that bring essential benefits [7, 8, 9]. Quantum
interconnect (QI) is a functional device to transfer quantum state from one
subsystem to another [10]. The core of a QI is a finely controlled light-
matter interface to correlate distant matters via flying photons [11, 12, 13,
14, 15, 16, 17]. Efficient QI among quantum emitters (QEs) has been explored
in different kinds of waveguide quantum electrodynamics (QED) systems [18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29].

Figure 1: Schematic diagram of a QI formed by a surface plasmon polariton
waveguide to entangle an array of $N$ QEs positioned at a distance $r_{a}$
from the axis of a metallic nanowire with radius $R$. The QEs are modeled as
two-level systems and equally separated by a distance $d$.

A surface plasmon polariton (SPP) supported by a metallic waveguide supplies
an ideal platform to realize QI. SPP is a hybrid excitation mode of
electromagnetic field (EMF) and electron density waves at the conductor-
dielectric interface. Confining EMF in a spatial scale far below the
diffraction limit, SPP has recently attracted an extensive interest to explore
strong quantized light-matter interactions [30, 31, 32, 33, 34, 35, 36, 37]
and led to fascinating applications in quantum technologies [38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48]. It has been found that SPP can act as a quantum
bus to mediate coherent interactions and generate quantum entanglement among
separated quantum emitters (QEs) [49, 50, 51, 52, 53, 54, 50, 51, 52, 55, 56,
57, 58], which exhibits an attractive prospect of realizing QI by SPP.
However, the metal absorption to the EMF makes the SPP short-lived, which
severely restricts its practical implementation of QI [59]. We really see that
the SPP-induced entanglement between QEs is dynamically transient and tends to
vanish in the long-time limit and with increasing the QE distance in the
absence of coherent driving [50, 51, 52, 55, 56, 57, 58]. In terms of
practical applications, one generally desires that a persistent entanglement
of the distant QEs could be mediated by QIs. Thus, how to suppress the
destructive influence of the lossy SPPs on the QEs is a key problem in
applying SPPs in QI.

Here, we propose a scheme of non-Markovian QI to distribute persistent
entanglement among an array of well separated QEs by the SPP propagating along
the surface of a metallic nanowire. We find that the subdiffraction feature of
the SPP endows our system with the strong light-matter coupling nature. It
causes that the $N$ QEs are so hybridized with the SPP that $N$ bound states
are formed between them, which invalidates the widely used Born-Markovian
approximation [49, 50, 51, 53, 54, 52, 51, 52, 57, 58]. This mechanism makes
the entanglement among the QEs mediated by the SPP persistently preserved in
the steady state. Solving the decoherence problem of the QEs caused by the
lossy SPP, our finding enriches our understanding on the light-matter
interactions in absorptive media. More importantly, such bound-state favored
realization of the QI of distant QEs paves the way for the applying the SPP in
designing scalable quantum devices, which is beneficial for various
applications in quantum technologies.

##  II Model and Hamiltonian

We consider a system consisting of an array of $N$ QEs coupled to a common
cylindrical metal nanowire (see Fig. 1). Being positioned on a line at a
distance $r_{a}$ from the axis of the nanowire, the QEs are equally separated
by a distance $d$ and modeled by a two-level system with resonant frequency
$\omega_{0}$. The electric permittivity of the nanowire is described by the
Drude model as
$\varepsilon_{m}(\omega)=\varepsilon_{\infty}-\omega_{p}^{2}/[\omega(\omega+i\gamma_{p})]$,
where $\varepsilon_{\infty}$ is the high-frequency limit of
$\varepsilon_{m}(\omega)$, $\omega_{p}$ is the bulk plasma frequency, and
$\gamma_{p}$ is a damping factor of the EMF in the metal.

Three distinct modes would be triggered by the radiation field emitted from
the QEs. The first one is the radiation mode to the free space. The second one
is the non-radiative mode absorbed by the metal. The last one is the SPP,
i.e., a hybrid mode of the EMF and the electron density waves, propagating
along the nanowire. The SPP enables a tight confinement of the EMF beyond the
diffraction limit on the metal-space interface, which makes the system an
ideal platform to explore and utilize the strong light-matter interactions. To
describe the quantum features of such interactions, a self-consistent
quantization scheme to the EMF in the absorptive medium is needed. A scheme
based on macroscopic quantum electrodynamics was developed using the Green’s
tensor method [60, 61], where the canonical commutation relations of the EMF
are guaranteed by introducing a Langevin noise to describe the medium
absorption. Then the quantized electric field is represented as

$\mathbf{\hat{E}}(\mathbf{r},\omega)=\frac{ic^{-2}\omega^{2}}{\sqrt{\pi\varepsilon_{0}/\hbar}}\int
d^{3}\mathbf{r}^{\prime}\sqrt{\text{Im}[\varepsilon_{\text{m}}(\omega)]}\mathbf{G(r},\mathbf{r}^{\prime},\omega)\cdot\mathbf{\hat{f}}(\mathbf{r}^{\prime},\omega),$

where $c$ is the speed of light, $\varepsilon_{0}$ is the vacuum permittivity,
and $\mathbf{\hat{f}}(\mathbf{r},\omega)$ obeying
$[\mathbf{\hat{f}(r},\omega),\mathbf{\hat{f}}^{{\dagger}}(\mathbf{r}^{\prime},\omega^{\prime})]=\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(\omega-\omega^{\prime})$
is the annihilation operator of EMF in the considered metal structure. The
Green’s tensor $\mathbf{G(r},\mathbf{r}^{\prime},\omega)$ characterizes the
field in frequency $\omega$ at $\mathbf{r}$ triggered by a point source at
$\mathbf{r}^{\prime}$ and is determined by the Maxwell-Helmholtz equation
$[{\boldsymbol{\nabla}}\times{\boldsymbol{\nabla}}\times-\omega^{2}c^{-2}\varepsilon_{m}\left(\omega\right)]\mathbf{G(r},\mathbf{r}^{\prime},\omega)=\mathbf{I}\delta(\mathbf{r}-\mathbf{r}^{\prime})$,
where $\mathbf{I}$ denotes an identity matrix. The spatial distribution of all
of the three modes has been incorporated in
$\mathbf{G(r},\mathbf{r}^{\prime},\omega)$. Thus, via solving the Helmholtz
equation, we obtain a complete quantized description of all the three modes
triggered by the EMF in the absorptive medium subject to the boundary
condition of the system geometry [62]. The Green’s tensor is analytically
solvable for a cylindrical nanowire [63].

Under the dipole and rotating-wave approximations, the Hamiltonian describing
the interactions between the QEs and the different modes triggered by the EMF
reads

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{l=1}^{N}\hbar\omega_{0}\hat{\sigma}^{{\dagger}}_{l}\hat{\sigma}_{l}+\int
d^{3}\mathbf{r}\int_{0}^{\infty}d\omega\hbar\omega\mathbf{\hat{f}}^{{\dagger}}(\mathbf{r},\omega)\cdot\mathbf{\hat{f}(r},\omega)$
(1)
$\displaystyle-\sum_{l=1}^{N}\int_{0}^{\infty}d\omega[\boldsymbol{\mu}_{l}\cdot\mathbf{\hat{E}}(\mathbf{r}_{l},\omega)\hat{\sigma}_{l}^{{\dagger}}+\textrm{H.c.}],$

where $\hat{\sigma}_{l}=|g_{l}\rangle\langle e_{l}|$ is the transition
operator from the excited state $|e_{l}\rangle$ to the ground state
$|g_{l}\rangle$, $\boldsymbol{\mu}_{l}$ is the dipole moment, and
$\mathbf{r}_{l}$ is the coordinate of the $l$th QE.

##  III Non-Markovian QI

It is easy to verify that the total excitation number
$\hat{\mathcal{N}}=\sum_{l}\hat{\sigma}_{l}^{{\dagger}}\hat{\sigma}_{l}+\int
d^{3}{\bf r}\int
d\omega\mathbf{\hat{f}}^{{\dagger}}(\mathbf{r},\omega)\cdot\mathbf{\hat{f}}(\mathbf{r},\omega)$
is conserved due to $[\hat{H},\hat{\mathcal{N}}]=0$. Thus, once being prepared
in a single-excitation state initially, the state of the total system evolves
to $|\Psi(t)\rangle=[\sum_{l}c_{l}(t)\hat{\sigma}_{l}^{{\dagger}}+\int
d^{3}\mathbf{r}\int d\omega
c_{\mathbf{r},\omega}(t)\mathbf{\hat{f}}^{{\dagger}}(\mathbf{r},\omega)]|G;\\{0_{\mathbf{r},\omega}\\}\rangle$,
where $|G\rangle$ denotes that all the QEs are in their ground state and
$|\\{0_{\mathbf{r},\omega}\\}\rangle$ is the vacuum state of the EMF. It can
be derived that the vector
$\mathbf{c}(t)=\left(\begin{array}[]{ccc}c_{1}(t)&\cdots&c_{N}(t)\\\
\end{array}\right)^{\text{T}}$ formed by the excited-state probability
amplitudes of the QEs obeys an integro-differential equation [64]

$\mathbf{\dot{c}}(t)+i\omega_{0}\mathbf{c}(t)+\int_{0}^{t}d\tau\int_{0}^{\infty}d\omega
e^{-i\omega(t-\tau)}\mathbf{J}(\omega)\mathbf{c}(\tau)=0,$ (2)

where $\mathbf{J}(\omega)$ is a $N$-by-$N$ matrix with element
$J_{lj}(\omega)=\omega^{2}\boldsymbol{\mu}_{l}\cdot\textrm{Im}[\mathbf{G}(\mathbf{r}_{l},\mathbf{r}_{j},\omega)]\cdot\boldsymbol{\mu}_{j}^{\ast}/(\pi\hbar\varepsilon_{0}c^{2})$
characterizing the correlated spectral density between the $l$th and $j$th
QEs. Such an correlation indicates that, although direct couplings between the
QEs are absent in Eq. (1), their indirect couplings are triggered via
exchanging the SPPs in the nanowire. For concreteness, we consider that all
the QEs have an identical dipole moment and are uniformly spaced. Such
symmetries make $J_{lj}(\omega)=J_{mn}(\omega)\equiv J_{|l-j|}(\omega)$ for
$|l-j|=|m-n|$. Reflecting the memory effects of the dynamics, the convolution
in Eq. (2) renders the dynamics non-Markovian. It is especially prominent in
quantum plasmonics because the light-matter interactions are significantly
enhanced by the sub-diffraction confinement of the SPPs to EMF [65, 66].

Figure 2: (a) Numbers of the bound states. (b) Evolution and (c) long-time
values of $\mathcal{C}_{2}(t)$ in different $r_{a}$ when $d=5$ nm by
numerically solving Eq. (2). In (c), the blue lines and red points are from
Eq. (6) and Eq. (2), respectively, and the orange region covers the values of
$\mathcal{C}_{2}(\infty)$ during its persistent oscillation. (d) Energy
spectrum of the whole system, which consists of a continuous band and two
branches of bound states in the band gap. Eigenenergies of the bound states
and evolution of $\mathcal{C}_{2}(t)$ in different $d$ when
$r_{a}=0.012\lambda_{0}$ in (e) and (f) and $r_{a}=0.0126\lambda_{0}$ in (g)
and (h). The dot lines highlight the critical points in which the entanglement
is lost in the long-time limit. Other parameters are $\hbar\gamma_{0}=10^{-4}$
eV, $R=0.01\lambda_{0}$, and $N=2$.

Equation (2) is formally solvable by the Laplace transform
$\mathbf{\bar{c}}(s)=\int_{0}^{\infty}e^{-st}\mathbf{c}(t)dt$, which results
in
$\mathbf{\bar{c}}(s)=\big{[}s+i\omega_{0}+\int_{0}^{\infty}d\omega\frac{\mathbf{J}(\omega)}{s+i\omega}\big{]}^{-1}\mathbf{c}(0)$.
$\mathbf{c}(t)$ is obtained via the inverse Laplace transform to
$\mathbf{\bar{c}}(s)$, which can be done by finding the poles from
($\varpi=is$)

$y_{j}(\varpi)\equiv\omega_{0}-\int_{0}^{\infty}d\omega\frac{D_{j}(\omega)}{\omega-\varpi}=\varpi,$
(3)

with $D_{j}(\omega)$ being the $j$th eigenvalues of $\mathbf{J}(\omega)$. It
is interesting to find that the root $\varpi$ multiplied by $\hbar$ is just
the eigenenergy of Eq. (1) in the single-excitation subspace [63]. It
indicates that the dynamics of the system characterized by ${\bf c}(t)$ is
intrinsically determined by the features of the energy spectrum formed by
$\hbar\varpi$. Because $y_{j}(\varpi)$ is a decreasing function in the region
of $\varpi<0$, Eq. (3) has a discrete root $\varpi^{b}_{j}$ provided
$y_{j}(0)<0$. The eigenstate corresponding to such a discrete $\varpi^{b}_{j}$
is called a bound state, whose formation has profound consequences on the
dynamics of the QEs [64, 67]. It is expected that, depending on the system
parameters, $N$ bound states could be formed at most for our $N$-QE
configuration. In the region of $\varpi>0$, $y(\varpi)$ is ill-defined and
thus Eq. (3) has an infinite number of roots, which form a continuous energy
band. According to the Cauchy’s residue theorem, we have

$\mathbf{c}(t)=\mathbf{Z}(t)+\int_{0}^{\infty}\frac{d\varpi}{2\pi}[\mathbf{\bar{c}}(-i\varpi+0^{+})-\mathbf{\bar{c}}(-i\varpi-0^{+})]e^{-i\varpi
t},$ (4)

where
$\mathbf{Z}(t)=-i\sum_{j=1}^{M}$Res$[\mathbf{\bar{c}}(-i\varpi_{j}^{b})]e^{-i\varpi_{j}^{b}t}$,
with $\text{Res}[\cdot]$ being the residue contributed by the $j$th bound
state $\varpi_{j}^{b}$ and $M$ being the number of formed bound states. The
second term comes from the eigenstates of the energy band. Oscillating with
time in continuously changing frequencies, the second term tends to zero in
the long-time limit due to the out-of-phase interference. Thus, if the bound
state is absent, then $\lim_{t\rightarrow\infty}\mathbf{c}(t)={\bf 0}$
characterizes a complete dissipation, while if the bound states are formed,
then $\lim_{t\rightarrow\infty}\mathbf{c}(t)=\mathbf{Z}(t)$ implies a
dissipation suppression. This result inspires us to propose a scheme to
realize a QI by the metal nanowire. Such a QI acting as a quantum bus permits
us to distribute a stable quantum correlation among the QEs acting as nodes of
a quantum network in the non-Markovian dynamics due to the formation of the
bound states.

First, we consider $N=2$ under the initial condition
$|\Psi(0)\rangle=|e_{1}g_{2}\rangle$. We calculate [63]

$\mathbf{Z}(t)=\frac{1}{2}\sum_{j=1}^{M}K^{(j)}_{01}e^{-i\varpi_{j}^{b}t}\left(\begin{array}[]{cc}1&(-1)^{j+1}\\\
\end{array}\right)^{\text{T}},$ (5)

with $K_{mn}^{(j)}=\big{[}1+\int
d\omega\frac{J_{m}(\omega)+(-1)^{j+1}J_{n}(\omega)}{(\omega-\varpi_{j}^{b})^{2}}\big{]}^{-1}$.
The entanglement of the QEs in a state $\rho$ is measured by concurrence
$\mathcal{C}_{2}=\max{\\{0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\\}}$,
where $\\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\\}$ are the
eigenvalues of
$\rho(\hat{\sigma}_{y}\otimes\hat{\sigma}_{y})\rho^{\ast}(\hat{\sigma}_{y}\otimes\hat{\sigma}_{y})$
in decreasing order [68]. From above bound-state analysis, we obtain

$\mathcal{C}_{2}(\infty)=\left\\{\begin{array}[]{cc}0,&M=0\\\
2(K_{01}^{(1)})^{2},&M=1\\\
2|(K_{01}^{(1)})^{2}-(K_{01}^{(2)})^{2}+D(t)|&M=2\end{array},\right.$ (6)

where $D(t)=2iK^{(1)}_{01}K^{(2)}_{01}\sin[(\varpi_{1}^{b}-\varpi_{2}^{b})t]$.
It indicates that a stable or a persistently oscillating entanglement between
the QEs would be generated once one or two bound states are formed,
respectively. This is in sharp contrast to the previous Born-Markovian
approximate results, where the entanglement approaches zero exclusively [50,
51, 52, 55, 56, 57, 58]. Revealing a mechanism to overcome the detrimental
influence of the damping SPP in the absorptive medium on the generated
entanglement between the QEs, this bound-state favored long-time feature
endows our system with an ability to realize QI via the mediation role of the
metal nanowire. Note that Eq. (6) reveals that the entanglement is lost at the
degenerate point of the two bound states.

In experiments, the metal could be silver with $\hbar\omega_{p}=9$ eV,
$\varepsilon_{\infty}=5.7$, and $\hbar\gamma_{p}=0.1$ eV [69]. Figure 2(a)
shows the number of bound states via solving Eq. (3). We see that two bound
states at most are formed in the small-$r_{a}$ regime. The entanglement
evolution in Fig. 2(b) reveals that, besides the severe oscillations in the
transient dynamics characterizing the rapid energy exchange between the QEs
caused by the near-field enhancement of the SPPs [70, 71], rich entanglement
features in different $r_{a}$, e.g., complete decay, stable trapping, and
Rabi-like oscillation, are exhibited in the steady state for different
$r_{a}$. The steady-state entanglement equals exactly to the ones predicted by
Eq. (6) [see Fig. 2(c)]. The regions where $\mathcal{C}_{2}(\infty)$ displays
distinct features matches well with the ones of different numbers of bound
state formed in the energy spectrum [see Fig. 2(d)]. The result verifies our
expectation that an efficient QI between the QEs with persistent entanglement
is realized due to the formation of one or two bound states. Figures 2(e)-2(h)
show the results in different distance $d$ between the QEs. We see that a
stable entanglement is generated for $d$ in support of the formation of a
bound state for the relatively large QE-nanowire distance $r_{a}$ [see Figs.
2(e) and 2(f)]. With decreasing $r_{a}$, the QE-SPP coupling is so enhanced
that two bound states are formed even when the QEs are separated in a distance
as large as $0.3\lambda_{0}$. Then, a persistently oscillating entanglement
can be established except for the accidental-degenerate points [see Figs. 2(g)
and 2(h)]. Efficiently avoiding the dissipation induced by the absorbing
metal, such bound-state favored stable entanglement is useful in applying SPPs
for the realization of quantum network [39, 72].

Figure 3: (a) Evolution of $\mathcal{C}_{3}(t)$ in different $r_{a}$ by
numerically solving Eq. (2). (b) Energy spectrum and (c) numbers of the bound
state as a function of $r_{a}$ by solving Eq. (3). (d-f) Evolution of the bi-
partite entanglement $\mathcal{C}_{12},\mathcal{C}_{13}$ and
$\mathcal{C}_{23}$ in different $r_{a}$. Other parameters are the same as in
Fig. 2(b) except for $N=3$.

Our result can be scaled up to the cases with a large number of QEs. It is
straightforward to analytically derive from Eqs. (3) and (4) the stable non-
zero $\mathbf{Z}(t)$ when different numbers of bound states are formed [63].
We consider the $N=3$ case under the initial condition
$|\Psi(0)\rangle=|e_{1}g_{2}g_{3}\rangle$. The three-QE system exhibits a
genuine tripartite entanglement, which is quantified by
$\mathcal{C}_{3}=\\{\frac{1}{3}\sum_{j=1}^{6}[(\mathcal{C}_{j}^{12|3})^{2}+(\mathcal{C}_{j}^{31|2})^{2}+(\mathcal{C}_{j}^{23|1})^{2}]\\}^{1/2}$.
Here
$\mathcal{C}_{j}^{mn|l}=\max\\{0,\sqrt{\lambda_{j,1}^{mn|l}}-\sum_{q>1}\sqrt{\lambda_{j,q}^{mn|l}}\\}$
is the bipartite concurrence, where $\lambda_{j,q}^{mn|l}$ are the eigenvalues
$\tilde{\rho}=\rho(L_{j}^{mn}\otimes\hat{\sigma}_{y}^{l})\rho^{\ast}(L_{j}^{mn}\otimes\hat{\sigma}_{y}^{l})$
and $L_{j}^{mn}(j=1,2,\cdots,6)$ are the six generators of the group SO$(4)$
[73, 74]. Figure 3(a) shows the evolution of $\mathcal{C}_{3}(t)$ in different
$r_{a}$. Similar entanglement features as the $N=2$ case, e.g., complete
decay, stable trapping, and Rabi-like oscillation but with more frequencies
than the $N=2$ case, are present in the steady state. The regions of these
features match well with the ones with zero, one, and more bound states formed
in the energy spectrum [see Fig. 3(b)]. At most three bound states can be
formed for small $r_{a}$ [see Fig. 3(c)]. The regions in the presence of the
nonzero tripartite entanglement also support a non-vanishing entanglement of
any bi-party partition of the three QEs. The bipartite entanglement in Fig.
3(d-f) reveals that persistent entanglement between any pair of the QEs is
also generated with the formation of bound states. This result reveals the
scalability of our QI realized by the SPP.

##  IV Discusion and conclusions

Our finding is realizable in the state-of-the-art experimental technique. The
QEs could be the CdSe quantum dots or the J aggregate. Their coupling to the
SPPs in silver nanowire have been extensively studied [75, 76, 77, 76, 78].
The bound state and its distinguished role in the non-Markovian dynamics have
been experimentally observed in both photonic crystal and untracold-atom
system [79, 80, 81]. The signatures of the bound-state favored interactions
between QEs have been recently found in Weyl photonic crystals and topological
waveguide [82, 83, 84].

In summary, we have proposed a scheme of non-Markovian QI formed by a metal
nanowire. A mechanism to generate a persistent entanglement among the distant
QEs via the SPP supported by the metal nanowire is discovered. It is revealed
that as long as one or more QE-SPP bound states are formed, the entanglement
would exhibit a finite-value stabilization or a persistent Rabi-like
oscillation in the long-time limit, respectively. Overcoming the destructive
impact of the metal absorption to the SPPs on the SPP-mediated QE
entanglement, our result enriches our understanding on the light-matter
interactions in lossy medium and supplies an instruction to apply the SPP in
designing scalable quantum devices.

##  V Acknowledgments

This work was supported by the National Natural Science Foundation of China
(Grants No. 12074106, No. 12275109, No. 12205128, No. 12247101, and No.
11834005) and China Postdoctoral Science Foundation (Grants No. BX20220138 and
No. 2022M710063).

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# Quarkonia dissociation at finite magnetic field in the presence of momentum
anisotropy

Indrani Nilima<EMAIL_ADDRESS>Department of Physics, Institute of
Science, Banaras Hindu University (BHU), Varanasi, 221005, India  Mujeeb Hasan
<EMAIL_ADDRESS>Lords Institute of Engineering and Technology, Hyderabad-
500091, Telangana, India  B. K. Singh<EMAIL_ADDRESS>Discipline of Natural
Sciences, PDPM Indian Institute of InformationTechnology Design and
Manufacturing, Jabalpur, 482005, India  Department of Physics, Institute of
Science, Banaras Hindu University (BHU), Varanasi, 221005, India  Mohammad
Yousuf Jamal<EMAIL_ADDRESS>School of Physical Sciences, Indian
Institute of Technology Goa, Ponda-403401, Goa, India

###### Abstract

In this study, we investigate the potential of heavy quarkonia within a
magnetized hot QGP medium having finite momentum anisotropy. The phenomenon of
inverse magnetic catalysis is introduced into the system, influencing the
magnetic field-modified Debye mass and thereby altering the effective quark
masses. Concurrently, the impact of momentum anisotropy in the medium is
considered that influence the particle distribution in the medium. The thermal
decay width and dissociation temperature of quarkonium states, specifically
the $1S$ and $2S$ states of charmonium and bottomonium, are computed. Our
results reveal that both momentum anisotropy and the inverse magnetic
catalysis effects play a significant role in modifying the thermal decay width
and dissociation temperature of these heavy quarkonia states.

##  I Introduction

The investigation of heavy quarkonia represents a fundamental avenue for
probing the deconfining nature of strongly interacting matter emerging from
heavy ion collisions. The hypothesized phase, known as quark-gluon plasma
(QGP), resembling conditions post the early universe, is being explored
through ongoing ultra-relativistic heavy ion collision (URHIC) experiments at
the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC)
Kharzeev:2007jp ; Skokov:2009qp ; Shovkovy ; Muller:PRD89'2013 . These
collisions generate remarkably intense magnetic fields, reaching magnitudes
estimated at $eB\sim m_{\pi}^{2}\equiv 10^{18}$ Gauss at RHIC and $eB\sim
15m_{\pi}^{2}\equiv 1.5\times 10^{19}$ Gauss, owing to the high relative
velocities of spectator quarks in non-central events Kharzeev:2007jp ;
Skokov:2009qp . Initially, there was limited attention given to the impact of
magnetic fields on thermal QCD mediums due to the perceived short lifespan of
these fields in influencing heavy-ion collision phenomenology. However, recent
theoretical predictions suggest that a thermal medium might emerge
simultaneously with the production of the magnetic field, resulting in an
extended lifespan due to the medium’s finite electrical conductivity
Tuchin:2010gx ; McLerran:2013hla . While the field weakens over time, its
decay within the medium is slower than in the vacuum. This scenario motivates
a comprehensive re-examination of QGP phenomenology in the presence of strong
magnetic fields in conjunction with heavy quark phenomenology within an
anisotropic medium. Notably, the inherent translation invariance of space is
disrupted by the magnetic field, leading to an expected absence of anisotropy
in the potential concerning the field’s direction. In our approach, the
emergence of anisotropy in the potential within the coordinate space
originates from the manifested momentum anisotropy associated with the
direction of anisotropy in the distribution function.

Recent studies have examined the impact of magnetic fields on various QCD
phenomena Alford:2013jva ; Fukushima:2015wck ; Das:2016cwd ; Singh:2017nfa ;
Hasan:2017fmf ; Kurain:2019 ; Bandyopadhyay:2021zlm ; Nilima:2022tmz ,
uncovering phenomena such as magnetic catalysis, inverse magnetic catalysis,
axial magnetic effects, and chiral magnetic effects. These investigations
prompt a deeper exploration of the interplay between strong magnetic fields
and heavy quark phenomenology within an anisotropic medium. In continuation of
our recent work on the dissociation of the heavy quark potential in the
presence of finite magnetic fields Nilima:2022tmz , this study focuses on
introducing local momentum anisotropy to investigate the dissociation
temperature of heavy quarkonia. It incorporates the effects of inverse
magnetic catalysis through the magnetic field-modified Debye mass. Recent
lattice quantum chromodynamics (LQCD) calculations have suggested coexisting
magnetic catalysis and inverse magnetic catalysis at non-zero QCD vacuum
temperatures and magnetic fields Bali1 ; Bali2 . This work integrates the
crucial inverse magnetic catalysis effect derived from constituent quark
masses obtained through lattice QCD simulations. The primary goal here is to
explore the inverse magnetic catalysis (IMC) effect on the constituent quark
masses within the heavy quark potential in an anisotropic medium. To achieve
this, we establish the standard formalism of the heavy quark potential in a
magnetic field as a foundation for our investigation Agotiya:2008ie ;
Thakur:2012eb ; Thakur:2013nia ; Kakade:2015xua ; Agotiya:2016bqr ;
Jamal:2023ncn . Subsequently, we derive the real and imaginary components of
an anisotropic heavy quark potential in a magnetic field, yielding modified
binding energies and thermal widths of quarkonium states. Finally, employing
the magnetic field-modified heavy quark potential within an anisotropic
medium, we scrutinize the dissociation of charmonium and bottomonium states.

The paper is organized as follows. In section II, we will discuss the basic
formalism of our present work, which includes discussions about the real and
imaginary parts of the heavy quark potential, decay width, binding energy, and
the Debye screening mass considering IMC and momentum anisotropy effects. In
section III, we will show our results for the same, as well as find out the
dissociation temperatures and discuss their magnetic field dependence.
Finally, in section IV, we shall conclude the present work.

##  II Formalism

The quark-antiquark interaction in the quarkonia state is characterized by the
Cornell potential Eichten:1978tg ; Eichten:1979ms ; Sebastian:2022sga . This
potential comprises both Coulombic and string components, denoted as:

$V(r)=-\frac{\alpha}{r}+\sigma r~{}.$ (1)

Here, $r$ represents the effective radius of the heavy quark and antiquark,
$\alpha$ is the strong coupling constant given by
$({\alpha}=\frac{g_{s}^{2}C_{F}}{4\pi};C_{F}=4/3)$, and $\sigma$ is the string
tension. We introduce anisotropy at the particle phase space distribution
level in our formalism using the method outlined in Refs. Romatschke:2003ms ;
Carrington:2014bla ; Jamal:2017dqs ; Jamal:2020hpy ; Kumar:2017bja , we obtain
anisotropic distribution functions from isotropic ones by rescaling
(stretching and squeezing) in one direction in momentum space:

$f({\mathbf{k}})\rightarrow f_{\xi}({\mathbf{k}})=C_{\xi}~{}f(\sqrt{{\bf
k}^{2}+\xi({\bf k}\cdot{\bf\hat{n}})^{2}}),$ (2)

Here, $f({\mathbf{k}})$ is the isotropic distribution function of the
particle, and ${\mathbf{\hat{n}}}$ is a unit vector indicating the direction
of momentum anisotropy. The parameter $\xi$ quantifies the anisotropic
strength in the medium, describing the degree of stretching ($-1<\xi<0$, or
prolate form) and squeezing ($\xi>0$, or oblate form) in the ${\bf\hat{n}}$
direction. $C_{\xi}$ is the normalization constant that can be obtained either
by keeping particle number or the debye mass intact in both the medium.

The magnetic field-dependent Cornell potential in the presence of an
anisotropic hot QCD medium is obtained by dividing it with the dielectric
permittivity $\epsilon(k,T,eB,{\xi})$ in Fourier space. This permittivity
encapsulates the medium information such as temperature, anisotropy, and
magnetic field. The modified Cornell potential is thus obtained as,

${\widetilde{V}}(k,T,eB,{\xi})=\frac{{{V}}(k)}{\epsilon(k,T,eB,{\xi})}~{},$
(3)

and by taking the inverse Fourier transform, we have

$V(r,T,eB,{\xi})=\int\frac{d^{3}\mathbf{k}}{(2\pi)^{3/2}}(e^{i\mathbf{k}\cdot\mathbf{r}}-1)\widetilde{V}(k,T,eB,{\xi})~{}.$
(4)

The dielectric permittivity of the medium is associated with the longitudinal
part of the gluon self-energy schneider  and can be obtained in Fourier space
from the $\omega\to 0$ limit of the temporal component of the medium-dependent
effective gluon propagator ($\Delta^{00}$) as:

$\epsilon^{-1}({\bf k},T,eB,{\xi})=-\lim_{\omega\to
0}k^{2}\Delta^{00}(\omega,{\bf k},T,eB,{\xi}).$ (5)

To obtain the real part of dielectric permittivity, we derive the real part of
the propagator given as,

$\displaystyle Re[\Delta^{00}]({\omega=0,{\bf k},T,eB,{\xi}})$
$\displaystyle=$
$\displaystyle\frac{-1}{k^{2}+m_{D}^{2}(T,eB)}+\xi\Big{(}\frac{1}{3\\{k^{2}+m_{D}^{2}(T,eB)\\}}$
(6) $\displaystyle-$
$\displaystyle\frac{m_{D}^{2}(T,eB)(3\cos{2\theta_{n}}-1)}{6\\{k^{2}+m_{D}^{2}(T,eB)\\}^{2}}\Big{)}.$

In a similar fashion, for the imaginary part of dielectric permittivity, the
imaginary part of the propagator is determined as follows,

$\displaystyle\text{Im}[\Delta^{00}]({\omega=0,{\bf k},T,eB,{\xi}})=$
$\displaystyle\pi{m_{D}^{2}(T,eB)}T\bigg{(}\frac{-1}{k\\{k^{2}+m_{D}^{2}(T,eB)\\}}+\xi\Big{(}\frac{-1}{3k\\{k^{2}+m_{D}^{2}(T,eB)\\}^{2}}$
$\displaystyle+\frac{3\sin^{2}{\theta_{n}}}{4k\\{k^{2}+m_{D}^{2}(T,eB)\\}^{2}}-\frac{2m_{D}^{2}(T,eB)\big{(}3\sin^{2}({\theta_{n}})-1\big{)}}{3k\\{k^{2}+m_{D}^{2}(T,eB)\\}^{3}}\Big{)}\bigg{)},$
(7)

Here, $\theta_{n}$ represents the angle between ${\bf k}$ and $\bf{\hat{n}}$.
Additionally, $\hat{s}=rm_{D}$ and $m_{D}\equiv m_{D}(T,eB)$ is the Debye mass
Nilima:2022tmz , characterizing the IMC effects through the medium-dependent
constituent quark mass $M_{f}(T,eB)$. This mass is derived from lattice QCD-
predicted values for normalized quark condensates $\langle
q\bar{q}\rangle_{f}(T,eB)$ at finite $eB$ Bali1 ; Bali2 :

$\displaystyle M_{f}(T,eB)=M_{f}(T=0,eB=0)\times\langle
q{\bar{q}}\rangle_{f}(T,eB)+m_{f},$ (8)

where $m_{f}$ is the light quark bare mass. In the limit $T\gg m_{f}$, we have

$\displaystyle M_{f}(T,eB)\approx M_{f}(T=0,eB=0)\times\langle
q{\bar{q}}\rangle_{f}(T,eB).$ (9)

The dispersion relation, now modified to incorporate the effective constituent
mass in a magnetized medium, is expressed as:

$\bar{E}_{f}^{l}=\sqrt{k_{z}^{2}+2l|q_{f}eB|+M_{f}(T,eB)^{2}}.$ (10)

Finally, employing the lattice QCD-inspired modified dispersion relation from
Eq. (10), the Debye screening mass is evaluated as:

$\displaystyle m_{D}^{2}$ $\displaystyle=$ $\displaystyle
4\pi\alpha_{s}(T)T^{2}\frac{N_{c}}{3}+\sum_{f}\dfrac{4\pi\alpha_{s}(T)|q_{f}eB|}{\pi^{2}T}$
(11) $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dk_{z}\sum_{l=0}^{\infty}(2-\delta_{l0})~{}{f}^{l}_{q}(\bar{E}_{f}^{l})~{}\left(1-{f}^{l}_{q}(\bar{E}_{f}^{l})\right),$

where $\alpha_{s}(T)$ is the temperature-dependent one-loop running coupling
Haque:2018eph :

$\displaystyle
g_{s}^{2}(T)=4\pi\alpha_{s}(T)=\frac{24\pi^{2}}{\left(11N_{c}-2N_{f}\right)\ln\left(\frac{2\pi
T}{\Lambda_{\overline{\rm MS}}}\right)},$ (12)

with $\Lambda_{\overline{\rm MS}}$ representing the $\overline{\rm MS}$
renormalization scale. Next, considering the short-distance limit $\hat{s}\ll
1$ Agotiya:2016bqr ; V_mD1 ; V_mD2 , we obtained of the real and imaginary
parts of the potential given as,

$\displaystyle Re[V(\hat{s},T,eB,\xi)]$ $\displaystyle=$
$\displaystyle\frac{\hat{s}~{}\sigma}{m_{D}}\left(1+\frac{\xi}{3}\right)-\frac{\alpha~{}m_{D}}{\hat{s}}\bigg{(}1+\frac{\hat{s}^{2}}{2}+\xi\left(\frac{1}{3}+\frac{\hat{s}^{2}}{16}\left(\frac{1}{3}+\cos\left(2\theta_{r}\right)\right)\right)\bigg{)},$
(13)

$\displaystyle\text{Im}[V(\hat{s},T,eB,\theta_{r},\xi)]$ $\displaystyle=$
$\displaystyle\frac{\alpha~{}\hat{s}^{2}~{}T}{3}\Big{\\{}\frac{\xi}{60}(7-9\cos
2\theta_{r})-1\Big{\\}}\log\left(\frac{1}{\hat{s}}\right)$ (14)
$\displaystyle+$
$\displaystyle\frac{\hat{s}^{4}~{}\sigma~{}T}{m_{D}^{2}}\Big{\\{}\frac{\xi}{35}\left(\frac{1}{9}-\frac{1}{4}\cos
2\theta_{r}\right)-\frac{1}{30}\Big{\\}}\log\left(\frac{1}{\hat{s}}\right),$

where $\theta_{r}$ is the angle between $\bf{r}$ and $\bf{\hat{n}}$. With this
simplification, the information of $T$ and $eB$ in the heavy quark potential
enters solely through the Debye mass. Next, we discuss the binding energy and
thermal width of heavy quarkonia. For the binding energy calculation under
small anisotropy, we solve the Schrödinger equation using the real part of the
potential as done in Margotta:2011ta ; Strickland:2011aa ; Thakur:2013nia . It
is important to note that we favor the real part over the imaginary part due
to its substantial magnitude (clearly depicted in Figs.1 to 8 for different
sets of parameters). The real part of the binding energy ${Re}[BE]$ is
expressed as:

${\text{Re}[BE(T,eB)]}=\bigg{(}\frac{m_{Q}\sigma^{2}}{m_{D}^{4}n^{2}}+\alpha
m_{D}+\frac{\xi}{3}\Big{(}\alpha
m_{D}+\frac{3m_{Q}\sigma^{2}}{m_{D}^{4}n^{2}}\Big{)}\bigg{)},$ (15)

The imaginary part of the potential give rise to the thermal width ($\Gamma$).
For the various quarkonia state, employing the imaginary part of the
potential, $\text{Im}~{}V(\hat{s},T,B,\theta_{r},\xi)$, the thermal width is
computed as Singh:2023zxu ; Agotiya:2016bqr :

$\displaystyle\Gamma(T,eB)=-\int
d^{3}{\bf{r}}\,\left|\Psi({{r}})\right|^{2}{\rm{Im}}~{}\
V({\hat{s};T,B,\theta_{r},\xi})~{},$ (16)

It is essential to highlight that we choose the Coulombic wave function
$\Psi(r)$ due to its long-range Coulombic tail, dominant at higher
temperatures. We consider both the Coulombic wave functions for the ground
state ($1s$, $J/\psi$ and $\Upsilon$) and the first excited state ($2s$,
$\psi^{\prime}$ and $\Upsilon^{\prime}$), respectively given as:

$\displaystyle\Psi_{1s}(r)$ $\displaystyle=\frac{1}{\sqrt{\pi
B_{r}^{3}}}e^{-\frac{r}{B_{r}}},$ (17) $\displaystyle\Psi_{2s}(r)$
$\displaystyle=\frac{1}{2\sqrt{2\pi
B_{r}^{3}}}\left(2-\frac{r}{B_{r}}\right)e^{-\frac{r}{2B_{r}}},$ (18)

Where $B_{r}$=$2/(m_{Q}\alpha_{s})$ represents the Bohr radius of the
quarkonia system. Utilizing Eq.16, the thermal width for $1s$ and $2s$ states
appears as follows, respectively:

$\displaystyle\Gamma_{1s}(T,eB)$
$\displaystyle=T\bigg{(}\frac{4}{\alpha_{s}~{}m_{Q}^{2}}+\frac{12\sigma}{{\alpha_{s}}^{4}~{}m_{Q}^{4}}\bigg{)}\bigg{(}1-\frac{\xi}{6}\bigg{)}m_{D}^{2}\log\left(\frac{m_{D}}{\alpha_{s}~{}m_{Q}}\right),$
(19) $\displaystyle\Gamma_{2s}(T,eB)$
$\displaystyle=\frac{8~{}m_{D}^{2}T}{{\alpha_{s}}^{4}~{}m_{Q}^{4}}\Big{(}1-\frac{\xi}{6}\Big{)}\bigg{(}7{\alpha_{s}}^{3}m_{Q}^{2}+192~{}\sigma\bigg{)}\log\left(\frac{2~{}m_{D}}{{\alpha_{s}}~{}m_{Q}}\right).$
(20)

Now, having the binding energy and the thermal dissociation width for both
ground and first excited states of charmonium and bottomonium states (i.e.,
$J/\psi/\psi^{\prime}$ and $\Upsilon/\Upsilon^{\prime}$), respectively, we can
determine the dissociation temperature for the quarkonium states using the
criteria $\Gamma_{n}(T,eB)=2\times B.E_{n}(T,eB)$ Mocsy:2007jz  for $n=1$ and
$2$.

##  III Results

Figure 1: Variation of the real part of potential with separation distance $r$
between $Q\bar{Q}$ for three different values of the magnetic field at
$\theta_{r}$ =0 and with fixed temperature T=100 MeV (left) and T=200 MeV
(right) at $\xi=0.1$.

Figure 2: Variation of the real part of potential with separation distance $r$
between $Q\bar{Q}$ for three different values of the magnetic field at
$\theta_{r}$ =0 and with fixed temperature T=100 MeV (left) and T=200 MeV
(right) at $\xi=0.3$.

Figure 3: Variation of the real part of potential with separation distance $r$
between $Q\bar{Q}$ for three different values of the magnetic field at
$\theta_{r}=\pi/2$ and with fixed temperature T=100 MeV (left) and T=200 MeV
(right) at $\xi=0.1$.

Figure 4: Variation of the real part of potential with separation distance $r$
between $Q\bar{Q}$ for three different values of the magnetic field at
$\theta_{r}=\pi/2$ and with fixed temperature T=100 MeV (left) and T=200 MeV
(right) at $\xi=0.3$.

Figure 5: Variation of the imaginary part of potential with separation
distance $r$ between $Q\bar{Q}$ for various values of magnetic field at
$\theta_{r}$ =0 with temperature T=100 MeV (left) and T=200 MeV (right) at
$\xi=0.1$.

Figure 6: Variation of the imaginary part of potential with separation
distance $r$ between $Q\bar{Q}$ for various values of magnetic field at
$\theta_{r}$ =0 with temperature T=100 MeV (left) and T=200 MeV (right) at
$\xi=0.3$.

Figure 7: Variation of the imaginary part of potential with separation
distance $r$ between $Q\bar{Q}$ for various values of magnetic field at
$\theta_{r}=\pi/2$ with temperature T=100 MeV (left) and T=200 MeV (right) at
$\xi=0.1$.

Figure 8: Variation of the imaginary part of potential with separation
distance $r$ between $Q\bar{Q}$ for various values of magnetic field at
$\theta_{r}=\pi/2$ with temperature T=100 MeV (left) and T=200 MeV (right) at
$\xi=0.3$.

Figure 9: The variation of the wavefunction of different charmonia (left) and
bottomonium (right) states with respect to $r$ at T=200 MeV. The solid curve
represents the 1S state, and the dotted curve is for the 2S state.

Figure 10: The variation of the wavefunction with respect to $r$ for 1s state
(left) and 2s state (right) at T=200 MeV. The solid curve represents the
charmonium state, and the dotted curve shows the bottomonium state.

In this section, we present significant findings pertaining to the magnetic
field-dependent modified heavy quarkonia potential, incorporating the IMC-
based quark condensate and accounting for momentum anisotropy. The analysis
encompasses an exploration of the impact of weak anisotropy within the hot QCD
plasma under specific conditions, $\xi=0.1$ and $\xi=0.3$. The other
parameters are fixed as $N_{f}=2$ quark flavors and $N_{c}=3$ colors, with
$\Lambda_{\overline{\rm MS}}=176$ MeV Haque:2014rua , and a string tension
value of $\sigma=0.184$ GeV2 as indicated in Jamal:2018mog .

Figures 1 and 2 depict the real part of the potential as a function of the
separation distance ($r$) between the $Q\bar{Q}$ pair under different magnetic
field strengths ($eB=0,0.2$ GeV2, and $0.4$ GeV2). These results are showcased
for $\xi=0.1$ and $0.3$, specifically exploring the parallel case
($\theta_{r}=0$) across two temperatures: $T=100$ MeV (left panel) and $T=200$
MeV (right panel). Corresponding plots for the perpendicular case
($\theta_{r}=\pi/2$) are displayed in Figs. 3 and 4. For the case of
$\xi=0.3$, there is a discernible increment in the values compared to
$\xi=0.1$. In both parallel and perpendicular configurations ($\theta_{r}=0$
and $\theta_{r}=\pi/2$), the real potential initiates from a negative value
and sharply rises towards zero, exhibiting a more pronounced behavior at
$T=100$ MeV than at $T=200$ MeV for both anisotropy values. This observation
suggests a tendency for the potential to stabilize at lower values with
increasing temperature. Moreover, it is noteworthy that the screening effect
is more pronounced at higher temperatures ($T=200$ MeV) compared to lower
temperatures ($T=100$ MeV) for both $\xi=0.1$ and $0.3$, irrespective of the
orientation (parallel or perpendicular). Additionally, there’s a subtle
sensitivity to anisotropy, where a slightly larger separation is observed for
$\theta_{r}=\pi/2$ compared to $\theta_{r}=0$. Concurrently, the presence of
the magnetic field serves to diminish the magnitude of the real part of the
potential, evident across both anisotropy values.

Employing the identical parameters, the imaginary part of the potential is
illustrated in Figs. 5 and 6 for $\theta_{r}=0$, corresponding to $\xi=0.1$
and $\xi=0.3$ respectively. Similarly, Figs. 7 and 8 present the outcomes for
$\theta_{r}=\pi/2$ with $\xi=0.1$ and $\xi=0.3$. Across both orientations, the
imaginary component consistently amplifies as the magnetic field strength
escalates, whether parallel or perpendicular. In terms of magnitude, this
imaginary part exhibits greater prominence at higher temperatures ($T=200$
MeV) compared to lower temperatures ($T=100$ MeV). This difference arises due
to the dependency on temperature and the Debye mass associated with the
magnetic field. Notably, the distinction between the magnetic field-dependent
imaginary potentials is more conspicuous at elevated temperatures ($T=200$
MeV) than at lower temperatures ($T=100$ MeV), observed consistently for both
$\theta_{r}=0$ and $\theta_{r}=\pi/2$. Moreover, the presence of the magnetic
field suppresses the magnitude of the imaginary potential.

Comparing with our earlier work Nilima:2022tmz , where we previously explored
the variation of both real and imaginary parts of the potential concerning the
separation distance ($r$) between $Q\bar{Q}$ pairs for distinct magnetic field
values at temperatures $T=100$ MeV and $T=200$ MeV, excluding momentum
anisotropy in the medium, reveals intriguing insights. The exponential decay
with distance becomes notably more pronounced at higher temperatures ($T=200$
MeV) compared to lower temperatures ($T=100$ MeV). Similarly, transitioning
from zero to non-zero magnetic field values and increasing their magnitudes
accentuates this dominance in the exponential behavior.

Fig. ( 9) shows the variation of the radial part of the eigen wavefunction for
$1s$ and $2s$ states of charmonium (left panel) and bottomonium (right panel)
with respect to $r$. Noticeable differences are observed in the peak height
and peak width of $1s$ state ($J/\psi$ and $\psi^{\prime}$) compared to $2s$
state ($\Upsilon$, and $\Upsilon^{\prime}$) for both charmonium and
bottomonium. The stronger binding nature of the $1s$ state ($J/\psi$ and
$\Upsilon$) is evident compared to the $2s$ state ($\psi^{\prime}$, and
$\Upsilon^{\prime}$). Fig. ( 10) provides a comparison between the eigen
wavefunctions of $1s$ state ($J/\psi$ and $\Upsilon$) and $2s$ state
($\psi^{\prime}$ and $\Upsilon^{\prime}$). Consistent with the earlier
conclusion, we observe that the binding nature of the $1s$ state ($J/\psi$ and
$\Upsilon$) is stronger than that of the $2s$ state ($\psi^{\prime}$ and
$\Upsilon^{\prime}$).

In broad terms, the real and imaginary components of the potential undergo
modifications influenced by temperature ($T$) and external magnetic field
strength ($eB$) in the presence of momentum anisotropy. These results stem
from the intricate behavior of Lattice QCD (LQCD) based quark condensates,
indicating an enhanced screening effect and a tendency towards dissociation
with rising values of $T$ and $eB$.

This behavior is again due to the aforementioned $T$ and $eB$ dependence of
the Debye mass. So, grossly, we notice that real and imaginary parts of heavy
quark potential are modified with $T$ and $eB$ due to the non-trivial profile
of LQCD-based quark condensates, and these modifications suggest more
screening and dissociation with increasing of $T$ and $eB$. They also indicate
the possibilities of low dissociation temperatures due to the external
magnetic field, which we will explicitly see later.

Extending our analysis, we employ the thermal width expressed in Eqs.
(19)/(20), coupled with the binding energy detailed in Eq. (15), to determine
the dissociation temperature ($T_{d}$) as the point of their intersection. Our
findings regarding dissociation temperatures are summarized in Tables 1 and 2
for $\xi=0.1$ and $0.3$, respectively. Specifically, at $\xi=0.1$, the
$J/\psi$ state undergoes dissociation at $T=192$ MeV for $eB=0$, $T=123$ MeV
for $eB=0.2$ GeV2, and $T=95$ MeV for $eB=0.4$ GeV2. Conversely, the
$\Upsilon$ state experiences dissociation at $T=290$ MeV for $eB=0$, $T=217$
MeV for $eB=0.2$ GeV2, and $T=207$ MeV for $eB=0.4$ GeV2 in the $\xi=0.1$
case. Notably, the dissociation temperature is the lowest for the
$\Upsilon^{{}^{\prime}}$ ($2s$-state), whereas it is higher for the $\Upsilon$
($1s$-state) compared to $J/\psi$ ($1s$-state). This hierarchy persists across
all three different magnetic field strengths, i.e., $eB=0$, $eB=0.2$ GeV2, and
$eB=0.4$ GeV2. In all the plots, the intersection points of thermal width and
binding energy are found to be smaller for $\xi=0.1$ compared to the $\xi=0.3$
case. Thus, for $\xi=0.3$, we observe slightly higher values of the
dissociation temperature for the quarkonium states at $eB=0,0.2$, and $0.4$
GeV2 compared to the previous scenario.

State $\downarrow$ Field strength $\rightarrow$ |  eB=0 |  eB=0.2 GeV2 |  eB=0.4 GeV2
---|---|---|---
$J/\psi$ |  192 |  123 |  95
$\Upsilon$ |  290 |  217 |  207
$\Upsilon^{{}^{\prime}}$ |  180 |  111 |  88
Table 1: The dissociation temperature($T_{D}$) for the quarkonia states (in units of MeV), when $\Gamma$= 2BE at $\xi$ =0.1 State $\downarrow$ Field strength $\rightarrow$ |  eB=0 |  eB=0.2 GeV2 |  eB=0.4 GeV2
---|---|---|---
$J/\psi$ |  194 |  126 |  97
$\Upsilon$ |  297 |  222 |  213
$\Upsilon^{{}^{\prime}}$ |  186 |  113 |  89
Table 2: The dissociation temperature($T_{D}$) for the quarkonia states (in
units of MeV), when $\Gamma$= 2BE at $\xi$ =0.3

We can observe a slight increase in dissociation temperatures when there is no
magnetic field, compared to situations where a magnetic field is present for
both $\xi=0.1$ and $0.3$. As previously discussed, the table highlights that
$\Upsilon$ dissociates at a higher temperature than $J/\psi$ and
$\Upsilon^{{}^{\prime}}$. It is notable that the dissociation temperature of
quarkonia states diminishes with the increasing magnetic field. This trend
closely mirrors the reduction of the transition temperature due to the
magnetic field, linked to the IMC effect. In essence, the consistent pattern
of decreasing dissociation temperature with an increasing magnetic field
persists in this scenario. Similar behavior was observed with slightly
modified magnitude in our previous work Nilima:2022tmz  where we had
considered the isotropic medium.

##  IV Conclusions

In summary, our investigation focused on the dissociation temperatures of
charmonium and bottomonium states, considering momentum anisotropy in the
medium. We revisited the medium-modified heavy quark-antiquark potential in
the presence of a finite magnetic field, incorporating the temperature and
magnetic field-dependent Debye mass within the gluon propagator. This
inclusion allowed us to capture the essence of IMC through quark condensates
inspired by LQCD. Examining the effects of different magnetic field strengths
($eB=0,0.2$ GeV2, and $0.4$ GeV2), we explored the real and imaginary parts of
the potential at two temperatures, $T=100$ MeV and $T=200$ MeV for $\xi=0.1$
and $0.3$. Our findings revealed an increasing screening mass with higher
magnetic field strength, indicating enhanced screening at elevated
temperatures, $T=200$ MeV compared to lower temperatures, $T=100$ MeV, both
for parallel ($\theta_{r}=0$) and perpendicular ($\theta_{r}=\pi/2$)
orientations. Additionally, we observed an increment in the magnitude of the
imaginary part of the potential with separation distance ($r$) for higher
magnetic field values. Furthermore, for a given separation distance, the
imaginary part exhibited greater magnitude at higher temperatures, $T=200$
MeV, compared to lower temperatures, $T=100$ MeV. Notably, the impact of
anisotropy on the imaginary part was found to be less pronounced than on the
real part. The observation is similar for both anisotropies ($\xi=0.1$ and
$0.3$) considered.

Further insights were gained by studying the wavefunctions of $1s$ and $2s$
states for charmonium and bottomonium, revealing distinctions in peak height
and width that contribute to our understanding of the binding nature of these
states. Employing these wavefunctions, we obtained the binding energy and
thermal dissociation width for various quarkonia states, and by employing a
criterion where twice the binding energy equals the thermal dissociation
width, we determined the dissociation temperatures for those states. Notably,
we observed a decrease in the dissociation temperature of quarkonia states
with an increase in magnetic field strength at fixed momentum anisotropy,
$\xi=0.1$ and $\xi=0.3$. We also observed the slightly higher values of the
dissociation temperature at $\xi=0.3$ for the quarkonium states at different
magnetic field strengths as compared to $\xi=0.1$. Looking ahead, our future
work aims to extend the present study by incorporating viscous corrections and
examining the dissociation of heavy quarkonia in a hydrodynamically expanding
viscous QGP.

## Acknowledgement

We would like to thank Aritra Bandyopadhyay for helpful discussions. IN
acknowledges the Women Scientist Scheme A (WoS A) of the Department of Science
and Technology (DST) for funding with grant no. DST/WoS-A/PM-79/2021. M.Y.J.
would like to acknowledge the SERB-NPDF (National postdoctoral fellow) with
File No. PDF/2022/001551.

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