# Direct generation of two-pair frequency entanglement via dual periodic
poling in lithium niobate waveguides

Aakash Warke<EMAIL_ADDRESS>Krishna Thyagarajan
<EMAIL_ADDRESS>Department of Physics, Bennett University,
Greater Noida 201310, India

###### Abstract

In this paper, we address the generation of a two-pair frequency entangled
state using type-0 spontaneous parametric down-conversion process in a dual
periodically poled lithium niobate waveguide. We show that, by suitable domain
engineering with two periods of quasi-phase matching grating, it is possible
to achieve a frequency entangled state with two different pairs of
frequencies. Numerical simulations show that the output state can be maximally
entangled. We also perform numerical simulations to address other interesting
and useful entangled states which can be generated with the help of our
scheme. The proposed scheme can help create efficient photonic setups required
for quantum communication systems and can have various applications
considering the increasing interests of energy-time entanglement in quantum
information regime.

Quantum optics, Waveguide optics, Quantum communication, Integrated optics

##  I Introduction

The foundation of quantum mechanics relies on the possibility of superposition
and entanglement AEPRA1935 ; JYPRL2013 . These phenomena form the basis for
many applications originating from quantum mechanics such as quantum
computation, quantum communication and more PBJSP1980 ; RFIJTP1982 ; KLNat2001
; BBPRL1993 ; PPRA2000 ; CBSp1992 . Hence, the generation of entangled states
is of crucial importance to such applications and to understand the
foundations of quantum mechanics. The quantum communication schemes encompass
photon entanglement to perform distinct communication tasks. This is the
reason why entangled photon pairs have attracted so much attention over the
past years YLPRA2016 ; MKNat2020 ; PKPRL1995 ; PKPRA1999 ; YCNat2016 ;
FSOE2019 ; DONPJ2020 ; YWAPR2021 . The process of spontaneous parametric down-
conversion employs the use of $\chi^{(2)}$ nonlinearity in crystals such as
lithium niobate to down-convert incident light of certain frequency, namely
pump, into two lower frequencies, signal and idler. This is possible through
certain phase matching conditions which are achieved via periodic poling in
these crystals. Multiple periods allow the simultaneous existence of the down-
converted pairs of photons, which in turn help us generate entangled photon
pairs STOSA2002 ; SLOSA2013 . This is a probabilistic way of generating
entangled photons. They can exist in different degrees of freedom such as
polarization, spatial modes, frequency etc. and a lot of research goes in
exploring such quantum states, for it not only has applications but also helps
us in understanding quantum mechanics at a deeper level KTPRA2009 ; VKSPRA2020
; JLOL2012 ; TJEPRL2015 ; DBPRA2015 ; LOPRA2010 ; CBPRA2013 ; HKOL2015 ;
FMOE2016 . Generation of hybrid- or hyper-entangled photon pairs has also
attracted a lot of attention over the years due to its several applications in
quantum technologies HJNat2014 ; DBJOSA2018 ; FGPRR2020 ; JLJOSA2013 . The use
of waveguides for this purpose of generating entanglement gives an edge as it
leads to augmented nonlinear efficiencies due to longer interaction lengths
and tight confinement of interacting waves. The periodically poled lithium
niobate is an essential substrate as it offers the largest nonlinear
coefficient $d_{33}$, thus leading to highest nonlinearity and ultimately,
effective entangled photon pairs at the output KTPRA2009 . Due to such high
nonlinear coefficients, various schemes for generation of entangled photons
have been reported by taking periodically poled lithium niobate (PPLN) or
periodically poled potassium titanyl phosphate (PPKTP) waveguides into
consideration HJPRL2014 ; OAJO2016 ; JZPRL2020 ; ZHLOE2011 .

Quantum frequency combs form a particularly useful resource for parallel
quantum communication processes. These require discrete modes of frequencies.
One way to deal with frequency’s continuous behavior is to discretize it and
generate biphoton combs entangled in frequency modes MKNat2019 . These kinds
of states were first explored using the spontaneous parametric down-conversion
(SPDC) process in bulk systems. They have also been examined for the higher
dimensional case using SPDC YJPRL2003 ; ZXNat2015 ; RJQST2016 . Frequency
combs were also generated using integrated optical micro-resonators owing to
spontaneous four-wave mixing FMOE2016 . Frequency combs are characterized by
closely lying frequencies. High dimensional frequency entangled states have
been generated and exploited for coherent manipulation in Hydex and silicon
nitride micro-ring resonators MKNat2017 ; PIOE2018 . More recently, a method
was proposed to control the comb’s symmetry by combining the cavity’s spectral
filtering effect with the control of temporal delay between the photons
corresponding to a pair GMNPJ2020 . The reported results are undoubtedly of
prime interest to us, but these solutions are based on setups that require
demanding stabilization and strict control.

In this paper, we present a scheme to directly generate two-pair frequency
entanglement which is the fundamental criteria for generation of frequency
combs, in a titanium in-diffused periodically poled lithium niobate waveguide
via type-0 spontaneous parametric down-conversion process. We make this
possible by suitable engineering of the quasi-phase matching (QPM) grating for
two periods which satisfy both the SPDC processes simultaneously. One period
corresponds to down-conversion of the pump light to signal and idler photons
of certain frequencies whereas the other period corresponds to down-converting
the same pump into a different set of frequencies. The output state will be
thus entangled in two pairs of frequencies. It is then possible to have a beam
splitter which transmits the signal wavelengths and reflects the idler
wavelengths. Furthermore, it is also possible to separate all the four
wavelengths into four ports leading to path entangled outputs. Compared to the
case of frequency combs, here the entangled frequencies can be quite
different. This can be interesting in cases where one of the entangled photons
is used for atom-photon interaction while the other photon is sent over an
optical fiber channel.

We assume extraordinary polarization for the pump light, signal and idler
photons. This will allow us to access the $d_{33}$ element of lithium niobate
leading to an effective output entangled state as discussed earlier. We
discuss interesting implications of considering different polarizations for
this process later on in the paper. This scheme should be of great interest to
applications requiring frequency entangled photon pairs, for instance,
frequency-multiplexed optical quantum information processing, and can even be
realized by commercially available optical components.

The paper is arranged as follows. In Section II, we give a quantum mechanical
analysis of the two SPDC processes to describe the generation of a two-pair
frequency entangled state. In Section III we present the numerical results of
simulations carried out by modeling the structure of our considered titanium
in-diffused channel waveguide, addressing the possibility to generate a
maximally entangled state. This is done with the intention of providing the
practicality of this idea. We then discuss the issues pertaining to bandwidth
of the two coupled interaction processes. We also discuss different entangled
states that can be generated through our proposed scheme and perform numerical
simulations to provide their practicality. In Section IV, we give a brief
conclusion of this paper.

##  II Quantum Mechanical Analysis of SPDC for the two interaction processes

We consider the process of parametric down-conversion in a titanium in-
diffused channel waveguide consisting of a z-cut, x-propagating lithium
niobate as its substrate. This substrate is poled as per the functional
dependence of nonlinear coefficient $d_{33}$ to satisfy both the required QPM
conditions simultaneously. We assume all the interacting modes involved in the
process to be extraordinarily polarized. It is therefore a type-0 spontaneous
parametric down-conversion process. As light of wavelength $\lambda_{p}$ is
launched into the crystal, it down-converts to photon pairs of different
wavelengths but the QPM conditions decide which of them should be more
efficient at the output. The objective is to generate an entangled state given
by:

$\Ket{\Psi}=\int d\omega_{s1}C_{1}\Ket{\omega_{s1},\omega_{i1}}+\int
d\omega_{s2}C_{2}\Ket{\omega_{s2},\omega_{i2}}$ (1)

Here, $\omega_{s1}$, $\omega_{s2}$ represent the signal frequencies and
$\omega_{i1}$, $\omega_{i2}$ represent the idler frequencies of the two down-
converted pairs. $C_{1}$ and $C_{2}$ are the amplitudes of the respective
down-conversion processes. In the case of a maximally entangled state, these
two values need to be equal. One period generates the pair
$\Ket{\omega_{s1},\omega_{i1}}$, and the other period generates
$\Ket{\omega_{s2},\omega_{i2}}$ from the incident pump light. It is necessary
that these conversions follow energy conservation which is given by:

$\displaystyle\omega_{p}=\omega_{s1}+\omega_{i1}$
$\displaystyle\omega_{p}=\omega_{s2}+\omega_{i2}$ (2)

First, we consider the electric field profiles of each of the four modes,
derive the interaction Hamiltonian using the $\chi^{(2)}$ nonlinearity, and
then use this Hamiltonian to derive the output state KTPRA2009 . Power of the
incident pump light in this process is usually high and thus, its electric
field can be assumed to be represented by a classical field. The signal and
idler fields are quantized and are therefore represented by their respective
operators. The electric field distributions for the pump and four interacting
modes are thus given by:

$\vec{E}_{\omega_{p}}=\frac{1}{2}e_{p}(\vec{r})E_{p}\big{[}e{{}^{i(k_{p}x-\omega_{p}t)}}+e{{}^{-i(k_{p}x-\omega_{p}t)}}\big{]}\hat{z}$
(3)

$\displaystyle\hat{E}_{\omega_{m}}=i\int{d\omega_{m}e_{\omega_{m}}(\vec{r})\sqrt{\frac{\hbar\omega_{m}}{2\epsilon_{m}L}}}\big{(}\hat{a}_{\omega_{m}}e^{i(k(\omega_{m})x-\omega_{m}t)}$
$\displaystyle-\hat{a^{\dagger}}_{\omega_{m}}e^{-i(k(\omega_{m})x-\omega_{m}t)}\big{)}\hat{z}\hskip
17.0pt$ (4)

The subscript $p$ stands for denoting pump in Eq.(3). In Eq. (4),
$m\in\\{s1,i1,s2,i2\\}$ and $\omega_{m}$ corresponds to the four frequencies
taken into consideration for the four discrete modes.
$e_{p}(\vec{r}),e_{\omega_{m}}(\vec{r})$ represent the transverse dependence
of these modal fields corresponding to extraordinary polarization of the
considered pump and the four frequencies. The length of interaction is
represented by $L$ and $\epsilon_{m}$ represents the optical dielectric
permittivity of the medium. Annihilation and creation operators of frequencies
into their respective modes are represented by $\hat{a}_{\omega_{m}}$ and
$\hat{a^{\dagger}}_{\omega_{m}}$.
Quasi-phase matching (QPM) ensures that the entanglement between the four
modes which will encompass these four discrete frequencies is generated
efficiently. These QPM periods are given by $K_{1}$ and $K_{2}$:

$\displaystyle
K_{1}=\frac{2\pi}{\Lambda_{1}}=2\pi\Big{\\{}\frac{n_{p}}{\lambda_{p}}-\frac{n_{s1}}{\lambda_{s1}}-\frac{n_{i1}}{\lambda_{i1}}\Big{\\}}$
(5) $\displaystyle
K_{2}=\frac{2\pi}{\Lambda_{2}}=2\pi\Big{\\{}\frac{n_{p}}{\lambda_{p}}-\frac{n_{s2}}{\lambda_{s2}}-\frac{n_{i2}}{\lambda_{i2}}\Big{\\}}$
(6)

Here, $n_{p},n_{s1},n_{i1},n_{s2},n_{i2}$ represent the effective indices of
the modes with extraordinary polarization at wavelengths
$\lambda_{p},\lambda_{s1},\lambda_{i1},\lambda_{s2},\lambda_{i2}$ respectively
and are calculated using the Sellmeier’s equations DEJOSA1997 . Now, using the
nonlinear polarization tensor, the total electric field components along x, y
and z axes are given by:

$\displaystyle E_{1}=0\hskip 60.0pt$ $\displaystyle E_{2}=0\hskip 60.0pt$
$\displaystyle
E_{3}=E_{\omega_{p}}+E_{\omega_{s1}}+E_{\omega_{i1}}+E_{\omega_{s2}}+E_{\omega_{i2}}$
(7)

Figure 1: This figure depicts the waveguide geometry for generating the two-
pair frequency entangled state as shown in Eq. (1), using two SPDC interaction
processes that satisfy the QPM conditions mentioned in Eq. (5) and (6). The
optic axis of this lithium niobate substrate is along the z direction. The
electric field profiles $\vec{E}_{\omega_{p}}$, $\hat{E}_{\omega_{s1}}$,
$\hat{E}_{\omega_{i1}}$, $\hat{E}_{\omega_{s2}}$ and $\hat{E}_{\omega_{i2}}$
are all along the optic axis as they are extraordinarily polarized.

The subscript indices 1,2, and 3 refer to components along x, y, and z axes as
shown in Fig. 1. Since this happens to be a type-0 down-conversion process in
a lithium niobate crystal, the $d_{33}$ coefficient is involved for derivation
of the Hamiltonian JLPRA2011 . Taking the expressions for electric field
profiles given in Eq. (3) and (4), the interaction Hamiltonian is derived as
shown in Eq. (8) KTPRA2009 .

$\hat{H}_{int}=-4\epsilon_{0}\int{d_{33}\big{(}\vec{E}_{\omega_{p}}\hat{E}_{\omega_{s1}}\hat{E}_{\omega_{i1}}+\vec{E}_{\omega_{p}}\hat{E}_{\omega_{s2}}\hat{E}_{\omega_{i2}}\big{)}d^{3}r}$
(8)

We assume a dual periodic poling KTPRA2009  and replace $d_{33}$ in Eq. (8)’s
interaction Hamiltonian by:

$\bar{d}_{33}=\frac{-4d_{33}}{\pi^{2}}\big{(}e^{iK_{1}x}+e^{-iK_{1}x}-e^{iK_{2}x}-e^{-iK_{2}x}\big{)}$
(9)

Now, substituting the expressions for respective electric field profiles, and
applying rotating-wave approximation, we derive the interaction Hamiltonian
which is given as:

$\displaystyle\hat{H}_{int}=\int\int
d\omega_{s1}d\omega_{i1}C_{1}^{(1)}\big{(}\hat{a^{\dagger}}_{\omega_{s1}}\hat{a^{\dagger}}_{\omega_{i1}}e^{-i(\omega_{p}-\omega_{s1}-\omega_{i1})t}$
$\displaystyle+h.c.\big{)}\hskip 20.0pt$ $\displaystyle+\int\int
d\omega_{s2}d\omega_{i2}C_{2}^{(1)}\big{(}\hat{a^{\dagger}}_{\omega_{s2}}\hat{a^{\dagger}}_{\omega_{i2}}e^{-i(\omega_{p}-\omega_{s2}-\omega_{i2})t}$
$\displaystyle+h.c.\big{)}\hskip 20.0pt$ (10) $\displaystyle
C_{1}^{(1)}=-\Bigg{(}\frac{4d_{33}E_{p}\hbar
I_{1}\sqrt{\omega_{s1}\omega_{i1}}}{\pi^{2}n_{s1}n_{i1}}\Bigg{)}\exp{\Big{(}-\frac{i\Delta
k_{1}L}{2}}\Big{)}$ $\displaystyle sinc\Big{(}\frac{\Delta
k_{1}L}{2}\Big{)}\hskip 12.0pt$ (11) $\displaystyle
C_{1}^{(2)}=-\Bigg{(}\frac{4d_{33}E_{p}\hbar
I_{2}\sqrt{\omega_{s2}\omega_{i2}}}{\pi^{2}n_{s2}n_{i2}}\Bigg{)}\exp{\Big{(}-\frac{i\Delta
k_{2}L}{2}}\Big{)}$ $\displaystyle sinc\Big{(}\frac{\Delta
k_{2}L}{2}\Big{)}\hskip 12.0pt$ (12)

Here, h.c. refers to the Hermitian conjugate and $\Delta k_{1(2)}$ represents
the phase mismatch given by:

$\displaystyle\Delta
k_{1}=K_{1}-2\pi\big{\\{}\frac{n_{p}}{\lambda_{p}}-\frac{n_{s1}}{\lambda_{s1}}-\frac{n_{i1}}{\lambda_{i1}}\big{\\}}$
(13) $\displaystyle\Delta
k_{2}=K_{2}-2\pi\big{\\{}\frac{n_{p}}{\lambda_{p}}-\frac{n_{s2}}{\lambda_{s2}}-\frac{n_{i2}}{\lambda_{i2}}\big{\\}}$
(14)

In Eq. (11) and (12), $I_{1}$ and $I_{2}$ denote the overlap integrals which
are crucial in determining the overlap between the interacting modes, which
decide whether the state can be maximally entangled or not. We discuss more
about this in Section III. These overlap integrals are given by:

$\displaystyle I_{1}=\int\int
e_{p}(\vec{r})e_{s1}(\vec{r})e_{i1}(\vec{r})dydz$ $\displaystyle
I_{2}=\int\int e_{p}(\vec{r})e_{s2}(\vec{r})e_{i2}(\vec{r})dydz$ (15)

Using the Hamiltonian shown in Eq. (11), we calculate the output state as:

$\Ket{\Psi}_{out}=e^{-i\hat{H}_{int}t/\hbar}\Ket{0,0}$ (16)

Taking Eq. (16), using the energy conservation, and substituting the values of
respective parameters, we get the following output state:

$\displaystyle\Ket{\Psi}=\int d\omega_{s1}C_{1}\Ket{\omega_{s1},\omega_{i1}}$
$\displaystyle+\int d\omega_{s2}C_{2}\Ket{\omega_{s2},\omega_{i2}}$ (17)

where,

$\displaystyle C_{1}=-\frac{it}{\hbar}C_{1}^{(1)}$ $\displaystyle
C_{2}=-\frac{it}{\hbar}C_{1}^{(2)}$ (18)

As one can observe, the state $\Ket{\Psi}$ is non-separable and thus,
entangled. Therefore, we have achieved generating entanglement between two
pairs of different frequencies using the aforementioned geometry in Fig. 1 of
a titanium-indiffused channel waveguide with lithium niobate as its substrate.
Depending on the overlap integrals and effective indices of the interacting
modes of the four frequencies, we can conclude whether the state can be
maximally entangled. As mentioned earlier, it is possible to now use a beam
splitter for the purpose of separation by transmitting frequencies of the
first mode of each pair, and reflecting the frequencies corresponding to the
second mode. It is also possible to separate all the four frequencies into
four ports which leads to a path entangled output. In Section III, we consider
practical values of various parameters involved in the process and show that
by properly designing the geometry of the waveguide, it is indeed possible to
achieve a maximally entangled state.

##  III Numerical Simulations and Discussion

For numerical simulation, we consider a titanium in-diffused channel waveguide
and follow the procedure given in Ref. KTPRA2009 . We find the parameters
$\alpha_{ym}$ and $\alpha_{zm}$, $m\in\\{s1,i1,s2,i2\\}$, by maximizing the
expression for $n_{eff}$. Using these parameters, we derive the analytical
expressions for $e_{p}(\vec{r})$, $e_{\omega_{s1}}(\vec{r})$,
$e_{\omega_{i1}}(\vec{r})$, $e_{\omega_{s2}}(\vec{r})$ and
$e_{\omega_{i2}}(\vec{r})$. We then solve the two overlap integrals given in
(15). The value of $n_{eff}$ gives the effective extraordinary indices of the
propagating pump and four frequency modes along the waveguide. These indices
are then used to obtain the QPM periods. Combining all of this together, we
can calculate the factor which relates $C_{1}^{(1)}$ and $C_{1}^{(2)}$ given
in Eq. (11) and (12) by the following expression KTPRA2009 :

$\gamma=\frac{min(C_{1}^{(1)},C_{1}^{(2)})}{max(C_{1}^{(1)},C_{1}^{(2)})}$
(19)

$\gamma$ is referred to as the degree of entanglement. As it is observed,
$n_{eff}$ is a function of width and height/depth of the waveguide, which
means that the QPM period will also be a function of these parameters of the
waveguide. This implies that the degree of entanglement will depend on the
width and depth of the waveguide. Therefore, by considering different values
of width and depth, we can determine which set of periods is the most suitable
to make sure $\gamma$ approaches to 1, leading to a maximally entangled state.
The values of wavelengths we choose for the pump and four discrete modes are
represented in Table 1 along with their respective extraordinary waveguide
index variations.

$Mode$ |  |  | $\lambda(nm)$ |  |  | $\Delta n_{e}$ |
---|---|---|---|---|---|---|---
$\lambda_{p}$ |  |  | 519 |  |  | 0.0037 |
$\lambda_{s1}$ |  |  | 780 |  |  | 0.0030 |
$\lambda_{i1}$ |  |  | 1551.03 |  |  | 0.0025 |
$\lambda_{s2}$ |  |  | 775 |  |  | 0.0030 |
$\lambda_{i2}$ |  |  | 1571.19 |  |  | 0.0025 |
Table 1: This table represents the values of considered wavelengths corresponding to the pump and the four modes of the entangled state, and their extraordinary waveguide index variation because of titanium in-diffusion. $Depth(\mu m)$ |  |  | $Width(\mu m)$ |  |  | $\gamma$ |  |  | $\Lambda_{1}(\mu m)$ |  |  | $\Lambda_{2}(\mu m)$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$6.5$ |  |  | $6.5$ |  |  | 0.9351 |  |  | 6.7874 |  |  | 6.8228 |
$8$ |  |  | $8$ |  |  | 0.9750 |  |  | 6.7903 |  |  | 6.8251 |
$10$ |  |  | $10$ |  |  | 0.9817 |  |  | 6.7969 |  |  | 6.8316 |
$12$ |  |  | $12$ |  |  | 0.9842 |  |  | 6.8027 |  |  | 6.8374 |
Table 2: This table represents the variation of the degree of entanglement and
the two QPM periods for different depths and widths of the titanium in-
diffused waveguide with lithium niobate as its substrate for the entangled
state represented by Eq. (17). $\gamma$ approaches 1 for a maximally entangled
quantum state.

Table 2 refers to the data acquired after performing the numerical simulations
by taking into account the state given in Eq. (17). As inferred, it is
possible to vary the parameters of the waveguide such that the state is
maximally entangled. Fig 2 represents the spectrum corresponding to signal
wavelengths $\lambda_{s1}$ and $\lambda_{s2}$. It is observed that outputs are
non-overlapping and the bandwidth calculated for $\lambda_{s1}$ for its
respective down-conversion process is 1.467 nm whereas the bandwidth
calculated for $\lambda_{s2}$ with its corresponding spectra is 1.407 nm. Fig.
3 represents the spectra of the interacting modes corresponding to idler
wavelengths.

Figure 2: This figure represents the spectra corresponding to type-0 SPDC
processes to generate the entangled state represented in Eq. (17) for
considered signal wavelengths of $\lambda_{s1}$ and $\lambda_{s2}$, which are
780 nm and 775 nm respectively. For a centimeter long interaction ($L$ = 1 cm)
with a waveguide of $w=h=10\mu m$, the bandwidth of the process corresponding
to $\lambda_{s1}$ is found to be 1.467 nm and the bandwidth of the process
corresponding to $\lambda_{s2}$ is found to be 1.407 nm. Figure 3: This figure
represents the calculated spectra for the SPDC processes corresponding to
idler wavelengths, $\lambda_{i1}$, $\lambda_{i2}$, which are 1.55103 $\mu m$
and 1.5719 $\mu m$ respectively. The SPDC process corresponds to the
generation of entangled state represented in Eq. (17) with $w$ = $h$ = 10
$\mu$m and $L$ = 1 cm. Bandwidths of $\lambda_{i1}$ and $\lambda_{i2}$ are
found to be 5.799 nm and 5.788 nm respectively.

It is interesting to note that if one wants to choose between more than two,
say three pairs of frequencies, then three quasi-phase matching periods will
have to be satisfied with appropriate consideration of wavelengths and
waveguide parameters. However, the more pairs we look forward to choose from,
lower the efficiency would be. Hence, different waveguide geometries can be
used to obtain a favorable output, for instance, in this case, a three-
waveguide directional coupler could be feasible DBPRA2015 .

As mentioned before, we have considered extraordinary polarization for all the
frequencies involved in the process. If we are to consider the polarization of
the propagating waves in the medium for the geometry mentioned in Fig. 1,
interesting entangled states can be generated. For example, consider the state
in Eq. (20).

$\displaystyle\Ket{\Psi}_{1}=\int
d\omega_{s1o}C_{1}\Ket{\omega_{s1o},\omega_{i1e}}$ $\displaystyle+\int
d\omega_{s2e}C_{2}\Ket{\omega_{s2e},\omega_{i2o}}$ (20)

Here, the subscripts $o$, $e$ refer to ordinary and extraordinary
polarizations of the four interacting frequency modes respectively. Cross-
polarization implies that it will take a type-II down-conversion process to
generate this entangled state. Such an entangled state will help acquire
different frequencies, only by measuring once, and thus can have numerous
applications. In order to provide the practicality of such a state, we
formulate the QPM periods analogously, and then perform numerical simulation
following Ref. KTPRA2009 . We assume the pump to be of ordinary polarization
and evaluate the effective indices for wavelengths shown in Table 1.
Corresponding waveguide index variations for ordinary polarization are shown
in Ref.KTPRA2009 . Table 3 shows the results of the variation of degree of
entanglement and poling periods for different widths and depths of the
waveguide for the entangled state given in (20).

$Depth(\mu m)$ |  |  | $Width(\mu m)$ |  |  | $\gamma$ |  |  | $\Lambda_{1}(\mu m)$ |  |  | $\Lambda_{2}(\mu m)$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$12$ |  |  | $12$ |  |  | 0.9864 |  |  | 4.5755 |  |  | 3.6457 |
$10$ |  |  | $10$ |  |  | 0.9866 |  |  | 4.5726 |  |  | 3.6440 |
$8$ |  |  | $8$ |  |  | 0.9876 |  |  | 4.5691 |  |  | 3.6421 |
$6.5$ |  |  | $6.5$ |  |  | 0.9975 |  |  | 4.5672 |  |  | 3.6412 |
Table 3: This table represents the data acquired for variation of the degree
of entanglement and the two poling periods accounting different values of
depths and widths of the waveguide to generate the entangled state shown in
Eq. (20).

As inferred from Table 3, such an entangled state can be maximally entangled
for appropriate choice of depth and widths of the waveguide. Fig. 4 represents
the calculated spectra of the modes corresponding to signal wavelengths
$\lambda_{s1o}$, $\lambda_{s1e}$, where $o$, $e$ are ordinary and
extraordinary polarizations respectively. Similarly, Fig. 5 represents the
calculated spectra of the SPDC process for the generation of entangled state
represented in Eq. (20) corresponding to idler wavelength variation.

Figure 4: This figure represents the calculated spectra of type-II SPDC
processes for the generation of entangled state represented in Eq. (20). The
considered wavelengths, $\lambda_{s1o}$ and $\lambda_{s2e}$, are 780 nm and
775 nm respectively. For a centimeter long interaction ($L$ = 1 cm) with
$w=h=6.5\mu m$, the bandwidth of the process corresponding to $\lambda_{s1o}$
is 0.499 nm and the bandwidth of the process corresponding to $\lambda_{s2e}$
is 1.952 nm. There seems to be a slight overlap in the two plots, but it is
not a significant overlap as $\lambda_{s1o}-\Delta\lambda_{s1o}$ lies outside
the range of $\lambda_{s2e}+\Delta\lambda_{s2e}$ and hence, can be
differentiated experimentally KTPRA2009 . $\Delta\lambda_{s1o}$ and
$\Delta\lambda_{s2e}$ represent the bandwidths of the two interaction
processes. Figure 5: This figure represents the bandwidth calculations of the
SPDC processes corresponding to idler wavelengths, $\lambda_{i1e}$ and
$\lambda_{i2o}$ of 1.55103 $\mu m$ and 1.5719 $\mu m$ respectively to generate
the entangled state represented in Eq. (20). For a centimeter long interaction
($L$ = 1 cm) with $w=h=6.5\mu m$, bandwidth corresponding to $\lambda_{i1e}$
is found to be 1.972 nm and that of $\lambda_{i2o}$ is found to be 8.032 nm.
The slight overlap in this case is not significant due to the same reason
mentioned in the explanation of Fig. 4.

It will be interesting to know how the value of $\gamma$ varies for a type-I
down-conversion process. We leave this in context of future work, as to the
best of our knowledge, both these entangled states will have the same purpose
i.e. choosing a single-mode frequency as per requirement through only one
measurement, which we have already shown in context of Table 3 and Fig. 3, 4.
Frequency entanglement has attracted a lot of attention in the past years,
both in theory and experiments, which is why direct generation of such
entangled states with widely different frequencies will turn out to be
extremely useful to today’s experiments involving energy-time entanglement.

##  IV Conclusion

In this paper, we have proposed a dual periodically poled substrate to
generate a two-pair frequency entangled state using type-0 SPDC process in a
PPLN waveguide. We also address the possibility of the generated state to be
maximally entangled by performing numerical simulations, considering values
for different parameters of the waveguide. We then take into account the
polarization of the four modes, and using type-II SPDC process, perform
simulations to find out that the output entangled state is more effective and
interesting. Such schemes should be of prime interest to quantum optical
communication technologies and experiments involving integrated optical
circuits.

##  V Acknowledgments

Aakash Warke would like to thank Ms. Akshata Rao, MSRIT, Bangalore, for her
useful expertise of SkecthUp software that helped build the geometry of the
considered waveguide.

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Holographic dark energy through Kaniadakis entropy in non flat universe
Suresh Kumar Pa,1, Bramha Dutta Pandeyb,2, Umesh Kumar Sharma c,3, Pankajd,4

${}^{a,\>b,\>c}$ Department of Mathematics, Institute of Applied Sciences and
Humanities, GLA University Mathura-281406, Uttar Pradesh, India

${}^{a,\>b}$ IT Department (Math Section), University of Technology and
Applied Sciences-Salalah, Oman

d Department of Mathematics(BSH), Pranveer Singh Institute Of Technology,
Kanpur-209305, Uttar Pradesh, India

1E-mail<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
3E-mail<EMAIL_ADDRESS>
4E-mail<EMAIL_ADDRESS>
###### Abstract

By extending the standard holographic principle to a cosmological framework
and combining the non-flat condition with the Kaniadakis entropy, we construct
the non-flat Kaniadakis holographic dark energy model. The model employs
Kaniadakis parameter $K$ and a parameter $c$. Derivation of the differential
equation for KHDE density parameter to describe the evolutionary behavior of
the universe is obtained. Such a differential equation could explain both the
open as well as closed universe models. The classification based on matter and
dark energy (DE) dominated regimes show that the KHDE scenario may be used to
specify the Universe’s thermal history and that a quintom regime can be
encountered. For open and closed both the cases, we find the expressions for
the deceleration parameter and the equation of state (EoS) parameter. Also, by
varying the associated parameters, classical stability of the method is
established. On considering the curvature to be positive, the universe favors
the quintom behavior for substantially smaller values as opposed to the flat
condition, when only quintessence is attained for such $K$ values.
Additionally, we see a similar behavior while considering the curvature to be
negative for such $K$ values. Therefore, adding a little bit of spatial
geometry that isn’t flat to the KHDE enhances the phenomenology while
maintaining $K$ values at lower levels. To validate the model parameters, the
most recent $30\;H(z)$ dataset measurements, in the redshift range $0.07\leq
z\leq 1.965$ are utilized. In addition, the distance modulus measurement from
the current Union 2.1 data set of type Ia supernovae are employed.

Keywords: Holographic dark energy, Non-flat, Kaniadakis entropy
PACS: 98.80.Es, 95.36.+x, 98.80.Ck

##  1 Introduction

The majority view in contemporary cosmology holds that both the early and late
universes witnessed accelerated expansion, which is backed by a vast number of
cosmic observations [1, 2]. The first strategy is to continue using general
relativity to explain gravity theory while introducing new types of matter,
such as inflation [3, 4] or the idea of dark energy [5, 6]. The second route
involves creating extended and modified gravitational theories, which, while
they generally provide an additional degree of freedom that might result in
acceleration, nevertheless have general relativity as a specific limit [7, 8,
9]. Mathematically, the Universe’s acceleration can alternatively be described
through holographic dark energy(HDE) [18, 19] and holographic inflation [10]
in an excellent way. But, technically speaking, it does not fit in the above
two solution approaches. However, the cosmic application [11, 12, 13] of the
holographic principle [14, 15, 16] allows for a different explanation of how
dark energy was created. The concept utilizes thermodynamics of black holes
and the relationship between a quantum field theory’s Ultraviolet cutoff and
largest theory’s distance. A quantum field theory must have an ultraviolet
cutoff for it to be applicable across large distances [17]. In particular, to
prevent the system from collapsing into a black hole, the total energy in a
particular system with volume-dependent entropy can not be more than an
equivalent-sized black hole’s mass whose area depends on entropy. When the
entire universe is seen as a system, holographic vacuum energy, or HDE with a
dynamical aspect, may be extracted [18, 19].

HDE’s cosmic effects turn out to be both fascinating [18, 19, 20, 21, 22, 23,
24] and consistent with observations [25, 26, 27, 28, 29, 30]. The fundamental
expression in the development of HDE is the one that relates a system’s
entropy to its radius. The Boltzmann-Gibbs entropy has been applied to black
holes and other cosmic phenomena to produce the most famous Bekenstein-Hawking
entropy. In contrast, the Boltzmann-Gibbs entropy’s one-parameter
generalization was proposed by Kaniadakis [31, 32]. This is the result of a
self-consistent and coherent relativistic statistical theory which maintains
the basic guidelines of normal statistical theory. Continuous deformation of
the original Maxwell-Boltzmann distribution function by one parameter is made
in such an expanded statistical theory. Thus, as a limited scenario, the usual
statistical theory is restored.
The holographic approach is used to produce HDE using the black hole entropy
expression. As a result, many versions of the theory may be generated by
varying the entropy. Applying Kaniadakis entropy in a black-hole framework
leads to

$S_{K}=\frac{1}{K}\sinh(KS_{BH}),$ (1)

where $S_{BH}$ is the Bekenstein-Hawking entropy. Bekenstein-Hawking entropy
is the limiting case of(1) i.e. $\lim_{K\rightarrow 0}S_{K}=S_{BH}$. $K<<1$
(especially $-1<K<1$) is expected for (1) to recover standard Bekenstein-
Hawking value. The basic Kaniadakis entropy can therefore be extended for
small $K$, yielding [33]

$S_{K}=S_{BH}+\frac{S_{BH}^{3}}{6}+\mathrm{O}(K^{4}). $ (2)

Various cosmological aspects of KHDE model, theoretically as well as
observationally in [34, 35, 36, 37, 38, 33, 39, 40, 41, 42, 43, 44, 45].

The combined study of the Planck Collaboration’s anisotropic power spectra of
the CMB and luminosity distance data indicates a non-flat universe at a $99\%$
confidence level [46]. Despite being much less than other energy components,
it is still conceivable that the spatial curvature makes a contribution to the
Friedmann equation. Studying a universe with a spatial curvature that is only
tangentially permitted by the inflation hypothesis, along with observations,
is therefore of more than just academic interest. In literature, Guo [22]
generalized the HDE model of M. Li [18, 19] by insisting non-flatness into
consideration. It was concluded that in order to maintain thermodynamics’
second law into picture, phantom-like situations will not exist. Setare [24]
studied the HDE model by considering the interaction among the constituent
sectors in a non flat universe. Sharma et al. [47] studied the KHDE model with
an apparent horizon in a non flat universe. Recently, Adhikari [48] studied
the Barrow HDE model with the largest theory’s distance as a future event
horizon to analyze the behavior of various cosmic parameters. The fractional
energy density due to curvature is taken as $\Omega_{k_{sc}}$.
$\Omega_{k_{sc}}>0$ signifies the spatially closed whereas $\Omega_{k_{sc}}<0$
corresponds to a spatially open universe. The Planck 2018 CMB temperature and
polarization anisotropy data favors the non flatness of the universe with
$\Omega_{k_{sc}}=-0.044_{-0.015}^{+0.018}$[49]. Additionally, by combining
$P18$ data with the full-shape (FS) universe power spectrum and adding the
$\Omega_{k_{sc}}=0.0023\pm 0.0028$ obtained from this collaboration together
with calculations from the BOSS DR12 CMASS example, one can attain results
with weaker constraints [50]. In any scenario, a considerable number of
scholars have been motivated to put tenable constraints on $\Omega_{k_{sc}}$
because the evolution of the universe is under the influence of spatial
curvature [46]. In light of the above discussions, our current study focuses
on constructing and analyzing the non-flat universe with Kaniadakis
holographic dark energy with future event horizon as IR cutoff.

The manuscript starts with a brief introduction. Next section is devoted to
formulating KHDE to describe the events of the closed and open Friedmann-
Robertson-Walker metric. We continue our extensive analysis of the cosmic
behavior in Section III by concentrating towards dark energy density, EoS and
deceleration parameters. Additionally, we’ll examine the squared sound speed
to confirm the model’s traditional stability. In Section IV, the techniques
used to analyze the data for this study are described. Last section is
dedicated to the summary of obtained results and conclusions.

##  2 Kaniadakis Holographic Dark Energy (KHDE) in Non-Flat Geometry

In this section, by assuming that spatial curvature is non-zero, we seek to
generate holographic dark energy.We pay special attention to a non-flat FRW
line element. Such a line element with $a(t)$ as scale factor is expressed by

$\mathrm{d}s^{2}=a(t)^{2}\left[\dfrac{\mathrm{d}r^{2}}{1-k_{sc}r^{2}}+r^{2}\mathrm{d}\Omega^{2}\right]-\mathrm{d}t^{2},$
(3)

where the choice $k_{sc}=-1,0,+1$ correspondingly result in open, flat, and
closed spatial curvatures.

Using (2) and the inequality $\rho_{d}L^{4}\leq S$ will give the expression
for the KHDE density $\rho_{d}$ given by [33, 51]

$\rho_{d}=\frac{3c^{2}M_{p}^{2}}{L^{2}}+K^{2}M_{p}^{6}L^{2},$ (4)

where $L,c$ and $K$ are length of horizon, model and Kaniadakis parameters
respectively and $M_{p}$ is the reduced Plank’s mass given by
$M_{p}=\dfrac{1}{\sqrt{8\pi G}}$. The standard Bakenstein-Hawking entropy is
recovered from Kaniadakis entropy in equation (4) for $K=0$.

To derive the basic results, we have considered the presence of only two
sectors in the universe namely: dark energy and dark matter. If $\rho_{m}$
represents the dark matter density and $p_{d}$ the KHDE pressure, the
corresponding Friedmann equations are given by

$3H^{2}+3\dfrac{k_{sc}}{a^{2}}=\rho_{d}+\rho_{m}, $ (5)
$2\dot{H}+3H^{2}+\dfrac{k_{sc}}{a^{2}}=-p_{d}, $ (6)

with Hubble parameter $H$ given by $H=\dfrac{\dot{a}}{a}$. The conservation
equation for the dark matter and KHDE sectors are given by

$\dot{\rho_{m}}+3H\rho_{m}=0,$ (7) $\dot{\rho_{d}}+3H(1+w_{d})\rho_{d}=0,$ (8)

with KHDE’s equation of state (EoS) parameter is $w_{d}$ defined by
$w_{d}=\dfrac{p_{d}}{\rho_{d}}$. For the sake of further analysis, the dark
matter, KHDE and curvature density parameters are defined respectively by
$\Omega_{m}=\dfrac{\rho_{m}}{3M_{p}^{2}H^{2}},\;\Omega_{d}=\dfrac{\rho_{d}}{3M_{p}^{2}H^{2}}$
and $\Omega_{k_{sc}}=\dfrac{k_{sc}}{a^{2}H^{2}}$. In the light of equation(5),
we get the relation among density parameters as

$\Omega_{d}+\Omega_{m}=1+\Omega_{k_{sc}}.$ (9)

The next stage is to define the theory’s longest length $L$, specifically the
holographic horizon, which is a part of the description of HDE. In the case of
the flat-universe, future event horizon $R_{h}$ will serve the purpose of the
theory’s largest distance $L$ to describe the accelerated expansion. But for
the non-flat universe model, a suitable modification is needed to get an
appreciable behavior of cosmic parameters for the accelerated expanding
universe[22, 24]. The suitably modified $L$ will be discussed in subsections
concerning closed and open spatial curvatures.

###  2.1 Case I: Closed Spatial Curvature ($k_{sc}=1$)

For the case of closed spatial curvature, the largest theory’s distance $L$ is
defined by $L=ar(t)$ [22, 24] with $r(t)$ defined by the relation

$\int_{0}^{r(t)}\dfrac{1}{\sqrt{1-k_{sc}r^{2}}}\mathrm{d}r=\int_{t}^{\infty}\dfrac{\mathrm{d}t}{a}=\dfrac{R_{h}}{a},$
(10)

which gives

$r(t)=\dfrac{1}{\sqrt{k_{sc}}}\sin{y},$ (11)

where

$y=\sqrt{k_{sc}}\dfrac{R_{h}}{a}=\sqrt{k_{sc}}\int_{x}^{\infty}\dfrac{\mathrm{d}x}{Ha},$
(12)

where $a=e^{x}$. We are interested to analyze the behavior of various density
parameters and hence will use expressions in terms of their present value as

$\Omega_{m}=\dfrac{\Omega_{m_{0}}H_{0}^{2}}{a^{3}H^{2}},\;\;\Omega_{k_{sc}}=\dfrac{\Omega_{{k_{sc}}_{0}}H_{0}^{2}}{a^{2}H^{2}},$
(13)

by considering

$\dfrac{\Omega_{k_{sc}}}{\Omega_{m}}=a\gamma,$ (14)

where $\gamma=\dfrac{\Omega_{{k_{sc}}_{0}}}{\Omega_{m_{0}}}$.

The universe’s curvature is related to the energy density and expansion rate
using the Friedmann equation (9) , given by [22]

$H=\dfrac{H_{0}\sqrt{\Omega_{m_{0}}}\sqrt{a^{-1}-\gamma}}{a\sqrt{1-\Omega_{d}}},$
(15)

Using equations (11), (12) and (15), we get

$L=\dfrac{a}{\sqrt{k_{sc}}}\sin\left[\sqrt{k_{sc}}\int_{x}^{\infty}\dfrac{1}{\sqrt{\Omega_{m_{0}}}H_{0}}\sqrt{\left(\dfrac{1-\Omega_{d}}{a^{-1}-\gamma}\right)}\mathrm{d}x\right].$
(16)

Another expression for $L$ is obtained through equations (3,4) and (15) given
by

$L=\sqrt{\dfrac{3H^{2}\Omega_{d}-\sqrt{9H^{4}\Omega_{d}^{2}-12c^{2}K^{2}M_{p}^{4}}}{2K^{2}M_{p}^{4}}}.$
(17)

As both the equations (16) and(17) are giving the value of $L$, we get

$\dfrac{a}{\sqrt{k_{sc}}}\sin\left[\sqrt{k_{sc}}\int_{x}^{\infty}\dfrac{1}{\sqrt{\Omega_{m_{0}}}H_{0}}\sqrt{\left(\dfrac{1-\Omega_{d}}{a^{-1}-\gamma}\right)}\mathrm{d}x\right]=\sqrt{\dfrac{3H^{2}\Omega_{d}-\sqrt{9H^{4}\Omega_{d}^{2}-12c^{2}K^{2}M_{p}^{4}}}{2K^{2}M_{p}^{4}}}.
$ (18)

Using the transformation $a=e^{x}$ in (18) and differentiating w.r.t. $x$ we
get

$\Omega_{d}^{\prime}=2\Omega_{d}(1-\Omega_{d})\left(\dfrac{B\sqrt{\Omega_{d}}}{A^{3/2}}\sqrt{\dfrac{A+B}{2c^{2}}-3k_{sc}e^{-2x}}-\dfrac{A+B}{A}+\dfrac{5-4e^{x}\gamma}{2-2e^{x}\gamma}\right),
$ (19)

where
$A=\dfrac{3e^{-3x}(1-e^{x}\gamma)H_{0}^{2}\Omega_{m_{0}}\Omega_{d}}{1-\Omega_{d}}$,
$B=\sqrt{A^{2}-12c^{2}K^{2}M_{p}^{4}}$.

The differential equation (19) describes the evolution of the KHDE by
supposing a closed universe with dark matter and dark energy as its primary
components. The case of $k_{sc}=0$ i.e. the curvature density
$\Omega_{k_{sc}}$ to be $0$ $(\gamma=0)$ corresponds to KHDE in flat universe
[33]. When $K\rightarrow 0$, we get $B=A$, leads the equation (19) to standard
HDE model of non-flat universe’s differential equation for the DE density
[22],
$\dfrac{\Omega_{d}^{\prime}}{\Omega_{d}^{2}}=(1-\Omega_{d})\left(\dfrac{1}{(1-a\gamma)\Omega_{d}}+\dfrac{2\cos
y}{c\sqrt{\Omega_{d}}}\right)$. We point out that when $K\rightarrow 0$ and
$\gamma=0$, it leads (19) in retrieving the equivalent differential equation
of the conventional HDE model for the flat universe. This equation is
$\Omega_{d}^{\prime}=\Omega_{d}(1-\Omega_{d})\left(1+2\sqrt{\dfrac{3M_{p}^{2}\Omega_{d}}{3c^{2}M_{p}^{2}}}\right)$,
and being independent of $x$, it can have an implicit analytic solution [18,
19].

We shall now examine how the KHDE’s EoS parameter $w_{d}=p_{d}/\rho_{d}$
behaves. Because of the conservation of matter sector, the Friedmann equations
(5-6) forces the conservation of dark energy sector, i.e.

$\dot{\rho_{d}}+3H\rho_{d}(1+w_{d})=0,.$ (20)

Differentiating (4), using (11), (12) with $\dot{L}=LH-\cos{y}$ and inserting
into (8), we get

$w_{d}=-1+\dfrac{2B}{3A}\left(1-\dfrac{1}{c}\sqrt{\Omega_{d}}\sqrt{\dfrac{A+B}{2A}}\cos{\left[ce^{-x}\sqrt{\dfrac{6k_{sc}}{A+B}}\right]}\right).$
(21)

Hence Obtaining $w_{d}$ requires knowledge of $\Omega_{d}$ from (19). On
considering $k_{sc}=0$, we achieve the EoS parameter expression obtained for
flat-universe [33]. When $K=0$ equation (21) becomes the EoS equation for non
flat universe $w_{d}=-\dfrac{1}{3}-\dfrac{2}{3c}\sqrt{\Omega_{d}}\cos y$ . For
$k_{sc}=K=0$, the relation (21) leads to the standard holographic dark energy
scenario $w_{d}=-\dfrac{1}{3}-\dfrac{2}{3c}\sqrt{\Omega_{d}}$ [18, 19].

From the Friedmann second equation (6) and Hubble parameter (15), the
deceleration parameter $q=-1-\dfrac{\dot{H}}{H^{2}}$ is deduced as

$q=\dfrac{1}{2}\left(1+3w_{d}\Omega_{d}+k_{sc}\dfrac{1-\Omega_{d}}{H_{0}^{2}\Omega_{m_{0}}\left(e^{-x}-\gamma\right)}\right).$
(22)

###  2.2 Case II: Open Spatial Curvature ($k_{sc}=-1$)

In this subsection the analysis is based on the negative spatial curvature
consideration, i.e. $k_{sc}=-1$. The largest theory’s distance $L$ is given by
$L=ar(t)$, where the expression for $r(t)$ can be obtained using the relation

$\int_{0}^{r(t)}\dfrac{1}{\sqrt{1+k_{sc}r^{2}}}\mathrm{d}r=\dfrac{R_{h}}{a},$
(23)

which gives the relation for $r(t)$ as

$r(t)=\dfrac{1}{\sqrt{|k_{sc}|}}\sinh{y},$ (24)

where, using the transformation $a=e^{x}$ we get

$y=\sqrt{|k_{sc}|}\dfrac{R_{h}}{a}=\sqrt{|k_{sc}|}\int_{x}^{\infty}\dfrac{1}{Ha}\mathrm{d}x.$
(25)

The calculation for $L$ will be performed similar to the closed spatial
curvature case and we get the relation

$\dfrac{a}{\sqrt{|k_{sc}|}}\sinh\left[\sqrt{|k_{sc}|}\int_{x}^{\infty}\dfrac{1}{\sqrt{\Omega_{m_{0}}}H_{0}}\sqrt{\left(\dfrac{1-\Omega_{d}}{a^{-1}-\gamma}\right)}\mathrm{d}x\right]=\sqrt{\dfrac{3H^{2}\Omega_{d}-\sqrt{9H^{4}\Omega_{d}^{2}-12c^{2}K^{2}M_{p}^{4}}}{2K^{2}M_{p}^{4}}}.
$ (26)

Taking the $x-$ derivative of the relation (26) by using the transformation
$a=e^{x}$ and applying equation (15) leads to the differential equation

$\Omega_{d}^{\prime}=2\Omega_{d}(1-\Omega_{d})\left(\dfrac{B\sqrt{\Omega_{d}}}{A^{3/2}}\sqrt{\dfrac{A+B}{2c^{2}}+3|k_{sc}|e^{-2x}}-\dfrac{A+B}{A}+\dfrac{5-4e^{x}\gamma}{2-2e^{x}\gamma}\right),
$ (27)

where
$A=\dfrac{3e^{-3x}(1-e^{x}\gamma)H_{0}^{2}\Omega_{m_{0}}\Omega_{d}}{1-\Omega_{d}}$,
$B=\sqrt{A^{2}-12c^{2}K^{2}M_{p}^{4}}$.

The development of KHDE for dust matter in an open universe is given by the
differential equation (27). The case of $k_{sc}=0$ leads to the KHDE for flat
universe [33, 51]. While as $K=0$, the current model tend towards the standard
holographic dark energy scenario for open universe [22],
$\dfrac{\Omega_{d}^{\prime}}{\Omega_{d}^{2}}=(1-\Omega_{d})\left(\dfrac{1}{(1-a\gamma)\Omega_{d}}+\dfrac{2\cosh
y}{c\sqrt{\Omega_{d}}} \right)$, using the relations $\cosh^{2}y-\sinh^{2}y=1
$ and $L=c\sqrt{\dfrac{3}{A}}=\dfrac{c}{H\sqrt{\Omega_{d}}}$. If we restrict
both, i.e. $k_{sc}=0$ and $K\rightarrow 0$, it returns the usual HDE in a flat
universe
$\Omega_{d}^{\prime}=\Omega_{d}(1-\Omega_{d})\left(1+2\sqrt{\dfrac{3M_{p}^{2}\Omega_{d}}{3c^{2}M_{p}^{2}}}\right)$
[18, 19].

Now we will obtain the EoS parameter expression for open spatial curvature in
a way exactly similar to the closed universe case. Taking the time derivative
of (4), using equations (8), (24) and (25) we get

$w_{d}=-1+\dfrac{2B}{3A}\left(1-\dfrac{1}{c}\sqrt{\Omega_{d}}\sqrt{\dfrac{A+B}{2A}}\cosh{\left[ce^{-x}\sqrt{\dfrac{6|k_{sc}|}{A+B}}\right]}\right).$
(28)

On considering $k_{sc}=0$, the EoS parameter expression for KHDE model with
flat universe is obtained [33, 51]. $K\rightarrow 0$ leads to the EoS
parameter expression for usual HDE model with open spatial curvature
$w_{d}=-\dfrac{1}{3}\left(1+\dfrac{2}{c}\sqrt{\Omega_{d}}\cosh y\right)$ .
Standard holographic dark energy model [18, 19] is recovered as a limiting
case of the current model, i.e. for $k_{sc}=0$ and $K\rightarrow 0$, we get
$w_{d}=-\dfrac{1}{3}\left(1+\dfrac{2}{c}\sqrt{\Omega_{d}}\right)$. The
deceleration parameter expression is provided using the same formula as in the
closed case.

$q=-1-\dfrac{\dot{H}}{H^{2}}=\dfrac{1}{2}\left(1+3w_{d}\Omega_{d}+k_{sc}\dfrac{1-\Omega_{d}}{H_{0}^{2}\Omega_{m_{0}}\left(e^{-x}-\gamma\right)}\right).$
(29)

In order to examine the KHDE model’s conventional stability, the squared sound
speed is stated as

$v_{s}^{2}=\dfrac{\mathrm{d}p_{d}}{\mathrm{d}\rho_{d}}=\dfrac{\rho_{d}}{\dot{\rho_{d}}}\dot{w_{d}}+w_{d}.
$ (30)

##  3 Cosmological Evolution

The KHDE model was developed in the part before this one with a future event
horizon by considering the spatial geometry to be open as well as closed.
Various equations pertaining to the evolutionary behavior of the universe such
as KHDE density, EoS and deceleration parameters were obtained. As a result,
we may now conduct a thorough analysis of the resultant cosmological behavior.
The equations (19) and (27) can be solved analytically only for $\gamma=0$ and
$K=0$. We must seek a numerical solution for the general case. Using the
straightforward equation $x=\ln{a}=-\ln{(1+z)}$, we may predict its behavior
in terms of the redshift $z$ as long as the solution for $\Omega_{d}$ is
available. Kaniadakis entropy, being an even function, restricts $K$ values to
be positive only.

Inline with observational data, equations (19) and (27) are solved numerically
by considering $\Omega_{d}(x=-\ln{(1+z)}=0)=0.7\equiv\Omega_{d_{0}}$,
$\Omega_{m}(x=-\ln{(1+z)}=0)=0.29\equiv\Omega_{m_{0}},$ and
$\Omega_{k}(x=-\ln{(1+z)}=0)=0.01\equiv\Omega_{k_{0}}$. By varying $c$ and
fixing $K$, figures 1 and 2 are plotted. Figure 1 explains the closed
universe’s thermal history whereas figure 2 characterizes the open universe’s
thermal history. Both the figures 1 and 2 imply that dark matter ruled the
whole universe in the past, partially KHDE dominated at present and fully KHDE
dominated in the near or far future.

(a)

(b)

(c)

Figure 1: The variation in the KHDE density parameter $\Omega_{d}$ with
redshift $z$ for the closed universe, in units when $M_{p}^{2}=1,\quad
H_{0}=67.9$. In order to be consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$.

(a)

(b)

(c)

Figure 2:  The variation in the KHDE density parameter $\Omega_{d}$ with
redshift $z$ for the open universe, in units when $M_{p}^{2}=1,\quad
H_{0}=67.9$. In order to be consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$.

The EoS parameter is plotted against redshift $z$ in figure 3 for closed
universe model and in 4 for open universe by assuming $K$ to be fixed and
varying $c$ . The KHDE model never crosses the line $w_{d}=-1$, shows pure
quintessence behavior for $c<1$, and $c=1$ results into the $\Lambda-$ CDM
model. For $c$ values greater than $1$, the EoS trajectories evolve in the
quintessence region, and by going across the divide line $w_{d}=-1$, in the
near or far future enters the phantom zone. As a result, when $c>1$, the KHDE
model acts like quintom.

(a)

(b)

(c)

Figure 3: The variation in the EoS parameter $w_{d}$ with redshift $z$ for the
closed universe, in units when $M_{p}^{2}=1,\quad H_{0}=67.9$. In order to be
consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$.

(a)

(b)

(c)

Figure 4:  The variation in the EoS parameter $w_{d}$ with redshift $z$ for
the open universe, in units when $M_{p}^{2}=1,\quad H_{0}=67.9$. In order to
be consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$.

Deceleration parameter plots w.r.t. $z$ are depicted in figures 5 and 6 for
closed and open universe models respectively by fixing the parameter $K$ and
varying $c$ values. Both the list of figures clearly state that the universe
started evolving in a decelerated phase. The phase shifted from deceleration
to acceleration at about $z\approx 0.6$. The future of the universe will
follow the accelerated phase irrespective of the nature of the model whether
closed or open.

(a)

(b)

(c)

Figure 5: The variation in the deceleration parameter $q$ with redshift $z$
for the closed universe, in units when $M_{p}^{2}=1,\quad H_{0}=67.9$. In
order to be consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$.

(a)

(b)

(c)

Figure 6: The variation in the deceleration parameter $q$ with redshift $z$
for the open universe, in units when $M_{p}^{2}=1,\quad H_{0}=67.9$. In order
to be consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$.

The squared sound speed against redshift is displayed to assist the model’s
conventional stability in figures 7 and 8 in the case of closed and open
universe models respectively by varying $c$ values and fixed $K$. Both figures
from 7 and 8 show that the KHDE model at hand is future stable for $c$ values
less than $1$ otherwise unstable irrespective of $K$ values.

(a)

(b)

(c)

Figure 7: The variation in the squared sound speed $v_{s}^{2}$ with redshift
$z$ for the closed universe, in units when $M_{p}^{2}=1,\quad H_{0}=67.9$. In
order to be consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$.

(a)

(b)

(c)

Figure 8:  The variation in the squared sound speed $v_{s}^{2}$ with redshift
$z$ for the open universe, in units when $M_{p}^{2}=1,\quad H_{0}=67.9$. In
order to be consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$.

##  4 Examination of Observed Data

This section discusses the non-flat KHDE model’s advanced research analysis. A
brief explanation is given of the datasets used in our research, with
particular emphasis on how the Type Ia Supernova and CC (cosmic chronometer)
mode were used to get the most recent Hubble parameter data (SNIa).

###  4.1 Observationally Obtained Hubble Dataset

We use the 30 H(z) observational dataset limitations from table 4 of [52],
which covers the range of redshifts from $0.07$ to $1.965$. The cosmic
chronometric (CC) method can be used to get this uncorrelated data. The
reasoning behind collecting these data is based on the possibility that the
Hubble dataset obtained via the CC approach is individualized. The method of
different galaxy dating determines the CC data of the universe that is
passively developing. Figures 9 and 10 demonstrate the development of $H(z)$
in the case of closed and open universe models respectively, and compares it
to the said $30$ points of $H(z)$ data. Clearly, the current model is
completely consistent with the observationally obtained $H(z)$ dataset.

(a)

(b)

(c)

Figure 9: The variation in the Hubble parameter $H$ with redshift $z$ for the
closed universe, in units when $M_{p}^{2}=1,\quad H_{0}=67.9$. In order to be
consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$. Bars represent the $30\;H(z)$
datapoints.

(a)

(b)

(c)

Figure 10:  The variation in the Hubble parameter $H$ with redshift $z$ for
the open universe, in units when $M_{p}^{2}=1,\quad H_{0}=67.9$. In order to
be consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$. Bars represent the $30\;H(z)$
datapoints.

###  4.2 Distance Modulus

The observations taken from the SNIa are extremely helpful for studying
cosmological models, especially as the main proof of an accelerating universe.
As a result, in addition to the CC data, we also used a sample of $580$ Union
2.1 scores mixed with SNIa from the distance modulus dataset [53]. The
redshift-luminosity distance connection is a well-known observational method
for studying the development of the universe [54]. The light travelling
towards a faraway luminescent body is redshifted as a result of the universe’s
expansion, and this causes us to be able to calculate the $z$-depependent
luminosity distance ($D_{L}$). We may calculate a source’s flux using the
luminosity distance, which is expressed by

$D_{L}=a_{0}r(1+z),$ (31)

with radial coordinate of the source $r$. Another definition of $D_{L}$ is
given by [5]

$D_{L}=\dfrac{c_{light}(1+z)}{H_{0}}\int_{0}^{z}\dfrac{1}{E(z)}\mathrm{d}z,$
(32)

with $E(z)=\dfrac{H(z)}{H_{0}}$. And hence the distance modulus is given by

$\mu=25+5\log_{10}\left(\dfrac{D_{L}}{M_{p}}\right).$ (33)

Equation(32) yields the relation for the distance modulus $\mu$ given by

$\mu=25+5\log_{10}\left[\dfrac{c_{light}(1+z)}{H_{0}}\int_{0}^{z}\dfrac{1}{E(z)}\mathrm{d}z\right].$
(34)

By combining the $580$ Union 2.1 [53] scores and the SNIa data , figures
LABEL:P31 to LABEL:P36 display the $z$-dependent period of distance modulus
$\mu(z)$ for the non-flat KHDE model, appropriately. The error graph of the
present model is presented by a solid line on togetherness of $580$ Union 2.1
results with SNIa datasets. Figures 11 and 12 show how the current model
significantly reflects the observed $\mu(z)$ values for each data point.

(a)

(b)

(c)

Figure 11:  The variation in the luminosity distance $\mu$ with redshift $z$
for the closed universe, in units when $M_{p}^{2}=1,\quad H_{0}=67.9$. In
order to be consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$. Bars display the $580$ Union
2.1 scores along with the SNIa results from the distance modulus data set.

(a)

(b)

(c)

Figure 12:  The variation in the luminosity distance $\mu$ with redshift $z$
for the open universe, in units when $M_{p}^{2}=1,\quad H_{0}=67.9$. In order
to be consistent with data, we now imposed
$\Omega_{d}(x=-\ln(1+z)=0)=\Omega_{d}[0]=0.7$. Bars display the $580$ Union
2.1 scores along with the SNIa results from the distance modulus data set.

##  5 Summary of Results

In this paper, we built the KHDE with future event horizon as IR cutoff in a
non-flat universe with model parameters $K$ and $c$. KHDE is constructed by
applying Kaniadakis entropy to a cosmological framework instead of normal
Bekenstein Hawking entropy and the holographic principle. We first offered a
straightforward differential equation for the holographic dark energy density
parameter $\Omega_{d}$ in order to examine the cosmic applications of KHDE.
Additionally, as a function of $\Omega_{d}$, we deduced an analytical
expression for the holographic dark energy equation-of-state parameter $w_{d}$
while accounting for both closed and open spatial geometry. Although the
aforementioned differential equation can be analytically solved in an implicit
form when $K=0$ and $\gamma=0$, in general case it can not be solved
analytically, hence one must numerically elaborate it.

A rich behavior is also displayed by the associated dark energy EoS parameter,
which can exhibit quintessence-like, or quintom like experience by crossing
the phantom-divide before or after the present. We then conducted a thorough
examination, demonstrating that the KHDE model may properly reflect the
universe’s thermal history, including the sequence of matter and dark energy
epochs. The change from deceleration to acceleration occurs in the vicinity of
$z\approx 0.6$ in agreement with observations, before it leads to a complete
dark energy dominance in the long term. The squared sound speed parameter
behavior shows that the KHDE model is stable for the present epoch but
unstable for the very beginning epoch. Furthermore, we tested the scenario
using observational data from the distance modulus dataset samples from $580$
Union 2.1 scores, and the combined SNIa data with 30 CC data points for
$H(z)$, to validate the values of the parameters $K$ and $c$ under
consideration, and found that the agreement is quite strong.

However, in order to more precisely constrain the new parameters, the present
analysis should be expanded with more data from the BAO (Baryon Acoustic
Oscillation), SNIa, and CMB probes. Further research on this topic is on the
way, and further studies will likely disclose more insights.

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