# Temporally-ultralong biphotons with a linewidth of 50 kHz

Yu-Sheng Wang,1 Kai-Bo Li,1 Chao-Feng Chang,1 Tan-Wen Lin,1 Jian-Qing Li,1
Shih-Si Hsiao,1 Jia-Mou Chen,1 Yi-Hua Lai,1 Ying-Cheng Chen,2,4 Yong-Fan
Chen,3,4 Chih-Sung Chuu,1,4 Ite A. Yu1,4,∗ 1Department of Physics, National
Tsing Hua University, Hsinchu 30013, Taiwan
2Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617,
Taiwan
3Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan
4Center for Quantum Technology, Hsinchu 30013, Taiwan
∗Corresponding author<EMAIL_ADDRESS>
###### Abstract

We report the generation of biphotons, with a temporal full width at the half
maximum (FWHM) of 13.4$\pm$0.3 $\mu$s and a spectral FWHM of 50$\pm$1 kHz, via
the process of spontaneous four-wave mixing. The temporal width is the
longest, and the spectral linewidth is the narrowest up to date. This is also
the first biphoton result that obtains a linewidth below 100 kHz, reaching a
new milestone. The very long biphoton wave packet has a signal-to-background
ratio of 3.4, which violates the Cauchy-Schwarz inequality for classical light
by 4.8 folds. Furthermore, we demonstrated a highly-tunable-linewidth biphoton
source and showed that while the biphoton source’s temporal and spectral width
were controllably varied by about 24 folds, its generation rate only changed
by less than 15%. A spectral brightness or generation rate per pump power per
linewidth of 1.2$\times$106 pairs/(s$\cdot$mW$\cdot$MHz) was achieved at the
temporal width of 13.4 $\mu$s. The above results were made possible by the low
decoherence rate and high optical depth of the experimental system, as well as
the nearly phase-mismatch-free scheme employed in the experiment. This work
has demonstrated a high-efficiency ultranarrow-linewidth biphoton source, and
has made a substantial advancement in the quantum technology utilizing
heralded single photons.

##  I Introduction

Sources of biphotons or pairs of time-correlated single photons are widely
utilized in quantum information research and applications. While one of the
paired photons is detected to trigger, or start, a quantum operation, the
other will be employed in the operation as a heralded single photon or, if
carrying a wave function or quantum state, a heralded qubit. Biphotons are
produced mainly via two kinds of schemes: the spontaneous parametric down
conversion (SPDC) and the spontaneous four-wave mixing (SFWM). A SPDC biphoton
source is commonly made of a nonlinear crystal placed inside an optical cavity
SPDC.Cavity2 ; SPDC.Cavity5 ; SPDC.Cavity6 ; SPDC.Cavity9 ; SPDC.Cavity10 ;
SPDC.Cavity11 ; SPDC.Cavity12 ; SPDC.Cavity13 ; SPDC.Cavity.Review ;
LongBiphotonSPDC.Cavity . A SFWM biphoton source generally consists of a cold
atom cloud SFWM.dLambda.ColdAtoms7.Du ; SFWM.dLambda.ColdAtoms9.Zhang ;
SFWM.dLambda.ColdAtoms10.Du ; SFWM.dLambda.ColdAtoms11 ;
SFWM.dLambda.ColdAtoms12 ; SFWM.dLambda.ColdAtoms13 ; SFWM.ladder.ColdAtoms2 ,
a hot atomic vapor SFWM.dLambda.HotAtom1 ; SFWM.dLambda.HotAtom2 ;
SFWM.dLambda.HotAtom5 ; SFWM.dLambda.HotAtom6 ; OurOpEx2021 ;
SFWM.dLambda.HotAtom7 ; OurPRR2022 ; SFWM.ladder.HotAtom5 ;
SFWM.ladder.HotAtom7 ; SFWM.ladder.HotAtom8 ; SFWM.ladder.HotAtom9 , or an
integrated photonics device with a built-in micro-resonator or waveguide IPD7
; IPD8 ; IPD9 ; IPD10 ; IPD11 ; IPD12 ; IPD.Review2 ; IPD.Review3 ;
LongBiphotonIPD . Both the SPDC biphoton source and the micro-
resonator/waveguide SFWM biphoton source can achieve a high generation rate
SPDC.Cavity2 ; SPDC.Cavity5 ; SPDC.Cavity6 ; SPDC.Cavity9 ; SPDC.Cavity10 ;
SPDC.Cavity11 ; SPDC.Cavity12 ; SPDC.Cavity13 ; SPDC.Cavity.Review ;
LongBiphotonSPDC.Cavity ; IPD7 ; IPD8 ; IPD9 ; IPD10 ; IPD11 ; IPD12 ;
IPD.Review2 ; IPD.Review3 ; LongBiphotonIPD . On the other hand, the atomic
SFWM biphoton source, which utilizes the effect of electromagnetically induced
transparency (EIT), is able to accomplish a long correlation time between the
pair of single photons, i.e., a long temporal width as well as a narrow
spectral linewidth of the biphoton wave packet SFWM.dLambda.ColdAtoms7.Du ;
SFWM.dLambda.ColdAtoms9.Zhang ; SFWM.dLambda.ColdAtoms10.Du ;
SFWM.dLambda.ColdAtoms11 ; SFWM.dLambda.ColdAtoms12 ; SFWM.dLambda.ColdAtoms13
; SFWM.dLambda.HotAtom1 ; SFWM.dLambda.HotAtom2 ; SFWM.dLambda.HotAtom5 ;
SFWM.dLambda.HotAtom6 ; OurOpEx2021 ; SFWM.dLambda.HotAtom7 ; OurPRR2022 .

A quantum operation employing biphotons of a narrower linewidth or,
equivalently, a longer temporal width, can achieve a better efficiency. For
example, a narrower linewidth of input photons can result in a higher storage
efficiency for quantum memory QM_OurPRL2013 ; QM_OurPRL2018 ; QM1 ; QM2 ; QM3
; QM4 ; QM5 , a greater success rate for quantum phase gates QPG1 ; QPG2 ;
QPG3 ; QPG4 ; QPG5 ; QPG6 ; QPG7 ; QPG8 , and a larger conversion efficiency
for quantum frequency converters QFC1 ; QFC_OurPRR2021 ; QFC2 ; QFC3 , which
utilize resonant or quasi-resonant transition schemes. As another example,
while the entanglement swapping is a key process in the quantum repeater
protocol for long-distance quantum communication QC1 ; QC2 ; QC3 ; QC4 ; QC5 ,
the deterministic entanglement swapping with the fidelity of 79% has been
demonstrated with ion qubits DES2008 , which operated at a narrow-linewidth
transition. In addition, the EIT effect has recently been observed in
superconducting qubits or artificial atoms driven by narrow-linewidth
microwaves SQ1 ; SQ2 ; SQ3 . One can foresee that narrow-linewidth photonic
qubits, which are converted to and from microwave coherences based on the EIT
effect QM_OurPRL2013 ; QM_OurPRL2018 ; QM1 ; QM2 ; QM3 ; QM4 ; QM5 , will be
employed in a quantum network that links superconducting-qubit-based quantum
computers QN1 ; QN2 ; QN3 .

Table 1: Biphoton sources with spectral profiles of the full width at the half
maximum (FWHM) below 1 MHz.

Process |  Medium |  Type(a) |  Temporal FWHM ($\mu$s) |  Spectral FWHM (kHz) |  Reference
---|---|---|---|---|---
SPDC | nonlinear crystal | MM | 0.33(b) | 670(c) | SPDC.Cavity10
SPDC | nonlinear crystal | MM | 0.83(b) | 265(c) | LongBiphotonSPDC.Cavity
SFWM | cold atom cloud | SM | 1.7 | 430 | SFWM.dLambda.ColdAtoms7.Du
SFWM | cold atom cloud | SM | 2.1 | 380 | SFWM.dLambda.ColdAtoms9.Zhang
SFWM | cold atom cloud | SM | 2.9 | 250 | SFWM.dLambda.ColdAtoms10.Du
SFWM | cold atom cloud | SM | 13.4 | 50 | this work
SFWM | hot atomic vapor | SM | 0.66 | 320 | OurOpEx2021 ; OurPRR2022

(a)MM denotes multimode, and SM denotes single mode.

(b)The FWHM of the envelope formed by all peaks in a temporally comb-like
structure.

(c)The spectral FWHM of the envelope.

Five groups have reported the sources of biphotons with sub-MHz linewidths.
Utilizing nonlinear crystals in the cavity-assisted SPDC process, Rambach et.
al. showed that their biphotons had a linewidth of 430 kHz SPDC.Cavity10 , and
Liu et. al. demonstrated a biphoton source with a linewidth of 265 kHz
LongBiphotonSPDC.Cavity . Both sources produced multi-mode biphotons, and the
frequency modes of the biphotons spanned several hundreds of MHz. Although
SPDC biphoton sources can generate single-mode biphotons, currently all of
these single-mode biphotons have linewidths larger than 1 MHz, e.g., Refs.
SPDC.Cavity10 ; SPDC.Cavity11 ; SPDC.Cavity12 . Using laser-cooled atoms in
the EIT-based SFWM process, Han et. al. achieved a biphoton source with a
linewidth of 380 kHz SFWM.dLambda.ColdAtoms9.Zhang , and Zhao et. al. reported
that their biphotons had a linewidth of 430 kHz SFWM.dLambda.ColdAtoms7.Du  or
250 kHz SFWM.dLambda.ColdAtoms10.Du . Employing an atomic vapor heated to 38∘C
in the EIT-based SFWM process, Hsu et. al. generated biphotons with a
linewidth of 320 kHz OurOpEx2021 ; OurPRR2022 . Among all the biphoton sources
of integrated photonics devices, the narrowest linewidth was 92 MHz
LongBiphotonIPD . There has been no report on the biphoton source with a
linewidth below 100 kHz until now. Please see Table 1 for the list of all
biphoton sources with linewidths below 1 MHz.

Here, we report a cold-atom SFWM source of biphotons with a tunable temporal
width. While maintaining non-classicality, the biphotons had a temporal width
as long as 13.4 $\mu$s, or a spectral linewidth as narrow as 50 kHz. A large
optical depth (OD) as well as a negligible decoherence rate in the
experimental system enabled the propagation delay time of the EIT effect to
dominate the temporal profile of the biphoton wave packet. Furthermore, we
were able to tune the temporal or, equivalently, spectral width by about 24
folds, and at the same time the change in the generation rate of the biphoton
source was less than 15%. A generation rate per pump power per linewidth of
1.2$\times$106 pairs/(s$\cdot$mW$\cdot$MHz) was achieved at the temporal width
of 13.4 $\mu$s, due to the nearly phase-mismatch-free scheme employed in our
experiment. In quantum operations, this biphoton source can meet any harsh
requirements of linewidth.

##  II Experimental Setup

Figure 1:  (a) Relevant energy levels of 87Rb atoms and the transition diagram
in the experiment. Ground states $|1\rangle$ and $|2\rangle$ and excited
states $|3\rangle$ and $|4\rangle$ are the Zeeman states of
$|5S_{1/2},F=1,m=1\rangle$, $|5S_{1/2},F=2,m=1\rangle$,
$|5P_{3/2},F=2,m=2\rangle$, and $|5P_{1/2},F=2,m=0\rangle$, respectively. The
pump field was blue-detuned with a detuning $\Delta_{p}$ of 200 MHz. The
signal (probe) photons are also called the Stokes (anti-Stokes) photons. The
frequency difference between $|1\rangle$ and $|2\rangle$ is 6.8 GHz,
corresponding to a wavelength difference of merely 0.014 nm. (b) Schema of the
experimental setup. The angle separation between the pump and coupling
propagation directions was approximately 1∘. The 795-nm signal photons and the
780-nm coupling field (the 780-nm probe photons and the 795-nm pump field)
propagated in the same direction. PMF: polarization-maintained optical fiber,
M: mirror, PBS: polarizing beam splitter, QWP: quarter-wave plate, BF780
(BF795): 780 nm (795 nm) bandpass filter, PF: polarization filter, SMF:
single-mode optical fiber, EF: etalon filter, and SPCM: single-photon counting
module.

We carried out the exepriment in cold 87Rb atoms, which were produced by a
magneto-optical trap (MOT). Details of the cold atoms and the MOT can be found
in Ref. QFC_OurPRR2021 . Before each measurement of biphotons, we momentarily
performed the dark-MOT and optically pumped all population to the single
Zeeman state of $|5S_{1/2},F=1,m=1\rangle$, i.e., the state $|1\rangle$ in
Fig. 1(a). The processes of the dark-MOT and the optical pumping are very
similar to those described in our previous works of Refs. OurPRL2012 ;
QM_OurPRL2013 . At the end of the processes, the optical depth (OD) of the
system was 110$\pm$5 throughout this work.

The transition diagram of the spontaneous four-wave mixing (SFWM) process is
shown in Fig. 1(a). Since all population was optically pumped to $|1\rangle$,
only the four Zeeman states specified in the caption were relevant to the
experiment. The pump field had the $\sigma-$ polarization and drove the
transition from $|1\rangle$ to $|4\rangle$ with a blue detuning of 200 MHz.
The coupling field had the $\sigma+$ polarization and drove the transition
from $|2\rangle$ to $|3\rangle$ resonantly. In the SFWM process, a pair of
signal and probe photons were emitted by the transitions of
$|4\rangle\rightarrow|2\rangle$ and $|3\rangle\rightarrow|1\rangle$, and had
the polarizations of $\sigma_{-}$ and $\sigma_{+}$, respectively. The
frequencies of the pump field, signal photon, coupling field, and probe photon
maintain the four-photon resonance. Since the energy level of $|1\rangle$ is
lower than that of $|2\rangle$, the signal and probe photons are also called
the Stokes and anti-Stokes photons. The pump field and Stokes photons formed
the Raman transition scheme, and their wavelengths were about 795 nm. The
coupling field and anti-Stokes photons formed the electromagnetically-induced
transparency (EIT) transition scheme, and their wavelengths were about 780 nm.

The sketch of the experimental setup is depicted in Fig. 1(b). We merged the
pump and coupling fields with a polarizing beam splitter (PBS). The two fields
propagated in the directions with a small angle separation of about
$1^{\circ}$. After the PBS, a quarter-wave plate made the pump and coupling
fields become $\sigma-$ and $\sigma+$ polarized. Their beam profiles
completely overlapped with the cigar-shaped atom cloud QFC_OurPRR2021 . The
pump and coupling fields had the same $e^{-2}$ full widths of 2.0 mm. In this
work, we set the pump power to 56 $\mu$W, and varied the coupling power from
60 $\mu$W to 1.5 mW. The Rabi frequency of the pump field was estimated
according to the Gaussian beam width and the transition dipole matrix element
of $|5S_{1/2},F=1,m=1\rangle$ $\rightarrow$ $|5P_{1/2},F=1,m=0\rangle$. The
peak intensity of the 56 $\mu$W pump beam corresponds to the Rabi frequency of
0.32$\Gamma$. The determination of the coupling Rabi frequency is illustrated
in the next paragraph.

We experimentally determined the quoted parameters of coupling Rabi frequency
($\Omega_{c}$), optical depth or OD ($\alpha$), and decoherence rate
($\gamma$) QFC_OurPRR2021 ; OurPRA2019 . First, the EIT spectrum of a weak
probe field was measured, and the separation distance between two transmission
minima, i.e. the Autler-Townes splitting, determined $\Omega_{c}$. In the
spectrum measurement, the OD was intentionally reduced such that the two
minima can be clearly observed. The Rabi frequency of the 60 $\mu$W (or 1.5
mW) coupling beam corresponded to $\Omega_{c}$ of 0.4$\Gamma$ (or
2.0$\Gamma$). Knowing the value of $\Omega_{c}$, we next measured the delay
time of a short probe pulse to determine the $\alpha$ used in the experiment.
Finally, we used the values of $\Omega_{c}$ and $\alpha$ as well as measured
the peak transmission of a long probe pulse to determine $\gamma$. This input
probe pulse was long enough that its output is not affected by the EIT
bandwidth. Once $\Omega_{c}$, $\alpha$, and $\gamma$ were determined, we
further compared the short-pulse data with the theoretical predictions. The
good agreement between the experimental data and theoretical predictions
demonstrated that the experimentally determined parameters of $\Omega_{c}$,
$\alpha$, and $\gamma$ are convincing.

To effectively reduce the contributions of the coupling and pump fields to the
single-photon counting modules (SPCMs), we intentionally collected Stokes
photons along the coupling propagation direction, and anti-Stokes photons
along the pump propagation direction. The reduction scheme will be described
in the next two paragraphs, and it attenuated the coupling and pump fields by
147 dB and 157 dB, respectively. Such an arrangement resulted in a phase
mismatch, i.e.,
$L|(\vec{k}_{p}-\vec{k}_{s}+\vec{k}_{c}-\vec{k}_{as})\cdot\hat{z}|$, of about
0.23 rad, where $L$ is the medium length, $\vec{k}_{p}$, $\vec{k}_{c}$,
$\vec{k}_{s}$, and $\vec{k}_{as}$ are the wave vectors of the pump and
coupling fields and the Stokes and anti-Stokes photons, and $\hat{z}$ is the
unit vector of the Stokes or anti-Stokes propagation direction. This phase
mismatch of 0.23 rad is a very manageable downside, since it makes a
negligible reduction of the biphoton’s generation rate.

Since the anti-Stokes (Stokes) photons and the pump (coupling) field
propagated nearly in the same direction, we installed a polarization filter
and a 780-nm bandpass filter (two 795-nm bandpass filters) to prevent the pump
(coupling) field from entering the single-mode optical fiber (SMF), collecting
the anti-Stokes (Stokes) photons. Each of the polarization filters is the
combination of a quarter-wave plate, a half-wave plate, and a polarizing
beamsplitter, which reduced the pump (coupling) power by 49 dB (49 dB). The
bandpass filter(s) added an additional attenuation of 38 dB (40 dB). After
each SMF, we further employed two etalons in series (an etalon) to decrease
the pump (coupling) leakage by about 70 dB (58 dB). The total extinction ratio
of 157 dB (147 dB) reduced the pump (coupling) power of 56-$\mu$W (1.5-mW) to
merely zero (12) photons/s.

Due to the anti-Stokes (Stokes) photons and the coupling (pump) field having
the very similar wavelengths and the same polarization, the above-mentioned
polarization and bandpass filters cannot block the leakage of the coupling
(pump) field into the anti-Stokes (Stokes) SMF. Fortunately, the angle
separation of $1^{\circ}$ between the anti-Stokes and coupling (between the
Stokes and pump) propagation directions had already reduced the leakage
significantly. After the SMFs, the two etalons in series (the etalon) further
decreased the coupling (pump) leakage by about 60 dB (60 dB). The total
extinction ratio of 130 dB (118 dB), including the attenuations due to the
angle separation and etalon(s), reduced the coupling (pump) power of 1.5 mW
(56 $\mu$W) to about 600 (350) photons/s. Note that the anti-Stokes (Stokes)
SPCM (Excelitas SPCM-AQRH-13-FC) has a quantum efficiency of 0.55 (0.56) and a
dark count rate of 220 (140) counts/s.

The two etalons in series for the anti-Stokes photons (the etalon for the
Stokes photons) had a net linewidth of 46 MHz (80 MHz), and a peak
transmission of around 18% (30%). Taking into account of the attenuation due
to optics, SMF’s coupling ratio, two etalons’ total transmission, and SPCM’s
quantum efficiency, the overall collection efficiency of the anti-Stokes
(Stokes) photons was 7.7% (13%). Inside the atomic cloud, the Stokes photon
propagated at the light speed in vacuum, and the anti-Stokes photon was slow
light due to the EIT effect. The Stokes SPCM detected a photon first, and
triggered a time tagger (Fast ComTech MCS6A5T8). After some delay time, the
anti-Stokes SPCM detected another photon which was recorded as a coincidence
count by the time tagger.

##  III Theoretical Model

The time correlation function between the anti-Stokes and Stokes photons,
i.e., the biphoton wave packet, is given by SFWM.Theory

$\displaystyle G^{(2)}(\tau)$ $\displaystyle=$
$\displaystyle\left|\int_{-\infty}^{\infty}\frac{d\delta}{2\pi}e^{-i\delta\tau}\frac{\sqrt{k_{as}k_{s}}L}{2}\chi(\delta)\right.$
(1) $\displaystyle\times$ $\displaystyle\left.{\rm
sinc}\\!\left[\frac{k_{s}L}{4}\xi(\delta)\right]e^{i(k_{s}L/4)\xi(\delta)}\right|^{2},$

where $\tau$ is the delay time of detecting an anti-Stokes photon upon a
Stokes photon’s trigger, $\delta$ is the two-photon detuning between the
Stokes photon and pump field (or $-\delta$ is that between the anti-Stokes
photon and coupling field), $k_{as}$ and $k_{s}$ are the wave vectors of the
two photons, $L$ is the medium length, $\chi(\delta)$ is the cross-
susceptibility of the anti-Stokes photon induced by the Stokes photon, and
$\xi(\delta)$ is the self-susceptibility of the anti-Stokes photon. The
formulas of the cross-susceptibility and self-susceptibility are shown below:

$\displaystyle\frac{\sqrt{k_{as}k_{s}}L}{2}\chi(\delta)$ $\displaystyle=$
$\displaystyle\frac{\alpha\Gamma}{4}\frac{\Omega_{p}}{\Delta_{p}+i\Gamma/2}$
$\displaystyle\times$
$\displaystyle\frac{\Omega_{c}}{\Omega_{c}^{2}-4(\delta+i\gamma)(\delta+i\Gamma/2)},~{}~{}$
$\displaystyle\frac{k_{s}L}{4}\xi(\delta)$ $\displaystyle=$
$\displaystyle\frac{\alpha\Gamma}{2}\frac{\delta+i\gamma}{\Omega_{c}^{2}-4(\delta+i\gamma)(\delta+i\Gamma/2)},$
(3)

where $\alpha$ represents the OD of the entire atoms, $\Omega_{p}$ and
$\Omega_{c}$ are the Rabi frequencies of the pump and coupling fields,
$\Gamma=$ 2$\pi$$\times$6 MHz is the spontaneous decay rate of the excited
state, $\Delta_{p}$ is the detuning of the pump field, and $\gamma$ is the
dephasing rate of the ground-state coherence, i.e., the decoherence rate.
Since the difference between the spontaneous decay rates of the excited states
$|3\rangle$ and $|4\rangle$ is merely about 5% in our case, we neglect the
difference and set the two rates to $\Gamma$.

##  IV Results and Discussion

Figure 2:  Representative biphoton wave packets. Circles connected with black
lines are the data of two-photon coincidence counts. Red lines are the
theoretical predictions of $G^{(2)}(\tau)$ in Eq. (1). Green lines are the
results of four-point moving average on the data points. In each inset,
squares represent the discrete Fourier transform of the data and blue line is
the Fourier transform of the red line. (a) The coupling power was 1.5 mW, and
the pump power and detuning were 56 $\mu$W and +200 MHz. Red line corresponds
to $(\alpha,\Omega_{c},\gamma)$ = $(115,2.1\Gamma,4.0{\times}10^{-3}\Gamma)$
and $(\Omega_{p},\Delta_{p})$ = $(0.32\Gamma,33.3\Gamma)$. The biphoton wave
packet had a temporal full width at the half maximum (FWHM) of 0.57 $\mu$s, a
spectral FWHM of 1.20 MHz, and a signal-to-background ratio (SBR) of 40. (b)
The coupling power was reduced to 63 $\mu$W, while the pump power and detuning
were the same. Red line corresponds to $(\alpha,\Omega_{c},\gamma)$ =
$(110,0.42\Gamma,3{\times}10^{-4}\Gamma)$ and $(\Omega_{p},\Delta_{p})$ being
the same. The biphoton wave packet had a temporal FWHM of 13.4 $\mu$s, a
spectral FWHM of 50 kHz, and a SBR of 3.4.

We systematically measured the biphoton wave packets or two-photon time
correlation functions by varying the coupling powers. The representative data
taken at the highest and lowest coupling powers of 1.5 mW and 56 $\mu$W are
shown by the circles in Figs. 2(a) and 2(b), respectively. The subfigure (a)
[or (b)] is the result of 120,000 (or 105,000) measurements, and each
measurement had a time window of 240 $\mu$s. Since we switched off (and on)
the MOT during (and after) each measurement, the duty cycle of the biphoton
generation is 0.8%. The width of time bin is 6.4 ns in (a) or 51.2 ns in (b).
Considering the overall collection efficiencies of the Stokes and anti-Stokes
photons, the generation rates of this biphoton source at the coupling powers
of 1.5 mW and 56 $\mu$W were 3460$\pm$140 pairs/s and 3340$\pm$40 pairs/s,
respectively.

The red lines in Fig. 2 are calculated from the two-photon time correlation
function, i.e., $G^{(2)}(\tau)$ in Eq. (1). During the biphoton-generation
time window of 240 $\mu$s, the OD of the system gradually decayed to about 90%
of its initial value. We take the OD’s decay into account, and all the quoted
values of the OD are the average value within this 240 $\mu$s. Since the OD
varied merely 10% and the variation was nearly linear during the measurement,
we just use its average value in the evaluation of $G^{(2)}(\tau)$. In the
calculation, $\alpha$ and $\gamma$ are set to the values, which are determined
by the classical-light data of slow light, and the values of $\Omega_{p}$ and
$\Delta_{p}$ are also experimentally measured or determined. The value of
$\Omega_{c}$ is adjustable to get the best match of the biphoton’s temporal
width. Note that $\alpha$, $\Omega_{c}$, and $\gamma$ affect the profile of
the biphoton wave packet, but $\Omega_{p}$ and $\Delta_{p}$ do not. In Figs.
2(a) and 2(b), $\Omega_{c}=2.1$$\Gamma$ and 0.42$\Gamma$, respectively, while
their values determined by the Autler-Townes splitting in the EIT spectrum are
$\Omega_{c}=2.0$$\Gamma$ and 0.40$\Gamma$. The parameters of $\alpha$,
$\gamma$, $\Omega_{p}$ and $\Delta_{p}$ (or $\Omega_{c}$) used in
$G^{(2)}(\tau)$ were measured (or verified) by the methods without using the
biphoton data. Hence, the red lines are regarded as the theoretical
predictions.

The coupling field also drove the far-detuned transition of
$|5S_{1/2},F=2,m=1\rangle$ $\rightarrow$ $|5P_{3/2},F=3,m=2\rangle$, and the
transition caused the photon switching effect PS ; OurOPL2006 . Consequently,
the value of $\gamma$ was approximately linear to that of $\Omega_{c}^{2}$
QM_OurPRL2013 ; OurPRA88in2013 ; QM_OurPRL2018 , i.e.,

$\gamma=\gamma_{0}+a\frac{\Omega_{c}^{2}}{\Gamma^{2}},$ (4)

where $\gamma_{0}$ is the intrinsic decoherence rate of the experimental
system and $a$ is the proportionality due to the photon switching effect. The
value of $\gamma_{0}$ was 2$\times$$10^{-4}$$\Gamma$ or 2$\pi$$\times$1.2 kHz.
The uncertainty plus day-to-day fluctuation of the decoherence rate is the
larger one of $\pm$1$\times$$10^{-4}$$\Gamma$ and $\pm$20%.

The green lines in Fig. 2 are the results of four-point moving average, which
reduces fluctuation of the data points. Since the temporal full width at the
half maximum (FWHM) of 0.57 (or 13.4) $\mu$s as determined by the red line is
much longer than the time-bin width of 6.4 (or 51.2) ns, the moving average
affects the profile of the biphoton wave packet very little. The consistency
between the experimental data (the green lines) and the theoretical
predictions are satisfactory. We performed the discrete Fourier transform
(DFT) on the biphoton wave packets and the Fourier transform (FT) on the
theoretical predictions. The baseline or background count of the data was
removed before the DFT or FT. The representative spectra are shown in the
insets of Fig. 2.

The temporal width or spectral linewidth of the biphoton wave packet is mainly
determined by the propagation delay time in the experimental system of a large
OD and a negligible decoherence rate. This is exactly demonstrated by Fig. 3,
showing that the FWHM of the biphoton’s temporal profile and the corresponding
spectral FWHM as functions of $\Omega_{c}^{2}$. Under a large OD, the
propagation delay time, $\tau_{d}$ ($=\alpha\Gamma/\Omega_{c}^{2}$), is much
larger than the inverse of the EIT bandwidth, $\tau_{b}$
($=1/\Delta\omega_{\rm EIT}$ or $\sqrt{\alpha}\Gamma/\Omega_{c}^{2}$). Under a
negligible decoherence rate, the coherence time, $\tau_{c}$ ($=1/\gamma$), is
far greater than $\tau_{d}$. Using the condition of
$\tau_{c}\gg\tau_{d}\gg\tau_{b}$, one can show that the biphoton’s spectral
FWHM, $\Delta\omega/2\pi$, is given by SFWM.Theory ;
SFWM.dLambda.ColdAtoms9.Zhang

$\frac{\Delta\omega}{2\pi}\approx\frac{0.88}{\tau_{d}}=\frac{0.88\Gamma}{\alpha}\left(\frac{\Omega_{c}}{\Gamma}\right)^{2}.$
(5)

In Fig. 3, the blue circles are the experimental data of $\Delta\omega/2\pi$
versus $(\Omega_{c}/\Gamma)^{2}$, and the blue line is the best fit of a
linear function with zero interception. The initial OD during the biphoton-
generation time window, i.e., $\alpha_{0}$, was 110$\pm$5\. Using
$\alpha=110{\pm}5$ in Eq. (5), we obtain $\Delta\omega/2\pi$ = 300$\pm$10
kHz$\times$$(\Omega_{c}/\Gamma)^{2}$. The slope of the best fit in Fig. 3 is
280 kHz. Therefore, the experimental result is very close to the theoretical
expectation of a nearly ideal biphoton source with a negligible decoherence
rate.

Figure 3: Temporal FWHM (red circles) and spectral FWHM (blue circles) of the
biphoton wave packet as functions of $\Omega_{c}^{2}$. In the measurement,
$\alpha$ = 110$\pm$5\. Blue line is the best fit of a linear function with
zero interception, which determines $\Delta\omega/2\pi$ (the spectral FWHM) =
280 kHz$\times$$(\Omega_{c}/\Gamma)^{2}$. In theory, i.e., Eq. (5), it is
expected that $\Delta\omega/2\pi=$ 300 kHz$\times$$(\Omega_{c}/\Gamma)^{2}$ at
$\alpha=110$.

The longest temporal or narrowest spectral FWHM in Fig. 3 is 13.4$\pm$0.3
$\mu$s or 50$\pm$1 kHz as demonstrated by Fig. 2(b). This is the best record
to date and also the first result of the biphoton linewidth below 100 kHz. The
very long biphoton had a signal-to-background ratio (SBR) of 3.4, showing that
the cross-correlation function between the Stokes and anti-Stokes photons,
$g^{(2)}_{s,as}(0)$, is 4.4 SFWM.dLambda.ColdAtoms10.Du . Both the auto-
correlation function of the Stokes photons, $g^{(2)}_{s,s}(0)$, and that of
the anti-Stokes photons, $g^{(2)}_{as,as}(0)$, are approximately equal to 2
OurPRR2022 ; AutoCorrelation . With $g^{(2)}_{s,as}(0)=4.4$ and
$g^{(2)}_{s,s}(0)=g^{(2)}_{as,as}(0)=2$, the Cauchy-Schwarz inequality for
classical light is violated by 4.8 folds, which clearly deomnstrates the
biphoton’s non-classicality. Furthermore, by increasing the coupling power we
were able to tune the temporal or spectral width to 0.57$\pm$0.02 $\mu$s or
1200$\pm$40 kHz as demonstrated by Fig. 2(a). While the temporal or spectral
width of the biphoton was shortened or enlarged by about 24 folds, its SBR was
significantly enhanced to 40.

Figure 4: Brightness, i.e., the generation rate per pump power, (red circles)
and spectral brightness, i.e., the brightness per linewidth, (blue circles) as
functions of $\Omega_{c}^{2}$. The pump power was kept at 56 $\mu$W, and the
generation rates of different coupling Rabi frequencies varied in the range of
3300$\sim$3800 pairs/s. During the entire measurement, the value of $\gamma$
was (3$\sim$40)$\times$$10^{-4}$$\Gamma$ corresponding to $\Omega_{c}=$
(0.4$\sim$2.0)$\Gamma$, and that of $\alpha$ was bewteen 105 and 115.
Considering the uncertainty or fluctuation in $\gamma$ of $\pm$20% and that in
$\alpha$ of $\pm$5, gray area represents the predictions based on Eq. (6).
Blue line is the best fit of the function $y=b/x$, where $b$ is the fitting
parameter.

We compared the generation rates of the biphoton source operating at different
coupling powers or equivalently producing various temporal widths. Under the
condition of a large OD and a negligible decoherence rate, i.e.,
$\Omega_{c}^{2}\gg 2\Delta\omega_{\rm EIT}\Gamma$ and $\Omega_{c}^{2}\gg
2\gamma\Gamma$, the generation rate is approximately proportional to

$\int_{-\infty}^{\infty}d\tau
G^{(2)}(\tau)\approx\frac{\alpha\Gamma}{2\pi}\frac{\Omega_{p}^{2}}{4\Delta_{p}^{2}+\Gamma^{2}}\exp\left(-\frac{\alpha\gamma\Gamma}{\Omega_{c}^{2}}\right).$
(6)

As long as the value of $\alpha\gamma\Gamma/\Omega_{c}^{2}$ is small, varying
the coupling power or tuning the biphoton temporal width changes the
generation rate a little. This is exactly the case in our experiment. Although
we changed the biphoton temporal width by about 24 folds, the generation rate
varied in the range of 3300$\sim$3800 pairs/s or fluctuated merely about
$\pm$7% as shown by Fig. 4. That is, we generated either a longer biphoton
pulse with a smaller amplitude or a shorter biphoton pulse with a larger
amplitude, while the area below the pulse or the pulse energy was
approximately the same OurPRA72in2005 . Only a negligible decoherence rate in
the EIT system can make it happen.

The brightness is defined as the generation rate per pump power, and the
spectral brightness is defined as the brightness per linewidth. Figure 4 shows
the brightness and the spectral brightness as functions of $\Omega_{c}^{2}$.
During the measurement, the pump power was kept the same, and thus the
behavior of the brightness is essentially the same as that of the generation
rate. We considered the fluctuations of the OD and the decoherence rate, and
utilized the predictions of Eq. (6) to plot the gray area. The consistency
between the data and the predictions is satisfactory. Furthermore, since the
brightness varied a little against $\Omega_{c}^{2}$ and the biphoton spectral
width is linearly proportional to $\Omega_{c}^{2}$, the spectral brightness or
generation rate per pump power per spectral linewidth approximately depends on
$1/\Omega_{c}^{2}$. Such a dependence is clearly seen in Fig. 4. In this work,
the highest spectral brightness is 1.2$\times$$10^{6}$
pairs/(s$\cdot$mW$\cdot$MHz).

##  V Conclusion

In conclusion, we were able to produce the narrow-linewidth biphoton with a
temporal FWHM of 13.4 $\mu$s, corresponding to a spectral FWHM of 50 kHz,
owing to a large optical depth and a negligible decoherence rate in the
experimental system. As for a wide range of the Rabi frequency or power of the
coupling field employed in this work, the large optical depth maintained the
criterion that the propagation delay time is far greater than the reciprocal
of the EIT bandwidth, and the negligible decoherence rate preserved the
criterion that the coherence time is much longer than the propagation delay
time. Consequently, not only the temporal profile of the biphoton wave packet
was predominately determined by the propagation delay time, but also we tuned
the temporal or spectral width by about 24 folds and, at the same time, kept
the generation rate approximately the same. This work has demonstrated a high-
efficiency ultranarrow-linewidth biphoton source, and achieved the milestone
of an ultralong biphoton wave packet.

## Acknowledgments

This work was supported by the Ministry of Science and Technology of Taiwan,
under Grants No. 109-2639-M-007-002-ASP and No. 110-2639-M-007-001-ASP.

## References

  * (1) X.-H. Bao, Y.Qian, J. Yang, H. Zhang, Z.-B. Chen, T. Yang, and J.-W. Pan, “Generation of narrow-band polarization-entangled photon pairs for atomic quantum memories,” Phys. Rev. Lett. 101, 190501 (2008).
  * (2) C.-S. Chuu, G. Y. Yin, and S. E. Harris, “A miniature ultrabright source of temporally long, narrowband biphotons,” Appl. Phys. Lett. 101, 051108 (2012).
  * (3) J. Fekete, D. Rieländer, M. Cristiani, and H. de Riedmatten, “Ultranarrow-band photon-pair source compatible with solid state quantum memories and telecommunication networks,” Phys. Rev. Lett. 110, 220502 (2013).
  * (4) M. Rambach, A. Nikolova, T. J. Weinhold, and A. G. White, “Sub-megahertz linewidth single photon source,” APL Photonics 1, 096101 (2016).
  * (5) D. Rieländer, A. Lenhard, M. Mazzera, and H. de Riedmatten, “Cavity enhanced telecom heralded single photons for spin-wave solid state quantum memories,” New J. Phys. 18, 123013 (2016).
  * (6) C.-H. Wu, T.-Y. Wu, Y.-C. Yeh, P.-H. Liu, C.-H. Chang, C.-K. Liu, T. Cheng, and C.-S. Chuu, “Bright single photons for light-matter interaction,” Phys. Rev. A 96, 023811 (2017).
  * (7) P.-J. Tsai, and Y.-C. Chen, “Ultrabright, narrow-band photon-pair source for atomic quantum memories,” Quantum Sci. Technol. 3, 034005 (2018).
  * (8) K. Niizeki, K. Ikeda, M. Zheng, X. Xie, K. Okamura, N. Takei, N. Namekata, S. Inoue, H. Kosaka, and T. Horikiri, “Ultrabright narrow-band telecom two-photon source for long-distance quantum communication,” Appl. Phys. Express 11, 042801 (2018).
  * (9) O. Slattery, L. Ma, K. Zong, and X. Tang, “Background and review of cavity-enhanced spontaneous parametric down-conversion,” J. Resv. Natl. Inst. Stan. 124, 124019 (2019).
  * (10) J. Liu, J. Liu, P. Yu, and G. Zhang, “Sub-megahertz narrow-band photon pairs at 606 nm for solid-state quantum memories,” APL Photon. 5, 066105 (2020).
  * (11) L. Zhao, X. Guo, C. Liu, Y. Sun, M. M. T. Loy, and S. Du, “Photon pairs with coherence time exceeding 1 $\mu$s,” Optica 1, 84-88 (2014).
  * (12) Z. Han, P. Qian, L. Zhou, J. F. Chen, and W. Zhang, “Coherence time limit of the biphotons generated in a dense cold atom cloud,” Sci. Rep. 5, 9126 (2015).
  * (13) L. Zhao, Y. Su, and S. Du, “Narrowband biphoton generation in the group delay regime,” Phys. Rev. A 93, 033815 (2016).
  * (14) P. Farrera, G. Heinze, B. Albrecht, M. Ho, M. Chávez, C. Teo, N. Sangouard, and H. de Riedmatten, “Generation of single photons with highly tunable wave shape from a cold atomic ensemble,” Nat. Commun. 7, 13556 (2016).
  * (15) C. Yang, Z. Gu, P. Chen, Z. Qin, J. F. Chen, and W. Zhang, “Tomography of the Temporal-Spectral State of Subnatural-Linewidth Single Photons from Atomic Ensembles,” Phys. Rev. Applied 10, 054011 (2018).
  * (16) R. Chinnarasu, C.-Y. Liu, Y.-F. Ding, C.-Y. Lee, T.-H. Hsieh, I. A. Yu, and C.-S. Chuu, “ Efficient generation of subnatural-linewidth biphotons by controlled quantum interference,” Phys. Rev. A 101, 063837 (2020).
  * (17) M. Steiner, V. Leong, M. A. Seidler, A. Cerè, and C. Kurtsiefer, “Photon bandwidth dependence of light-matter interaction,” Opt. Express 25, 6294-6301 (2017).
  * (18) C. Shu, P. Chen, T. K. A. Chow, L. Zhu, Y. Xiao, M. M. T. Loy, and S. Du, “Subnatural-linewidth biphotons from a Doppler-broadened hot atomic vapour cell,” Nat. Commun. 7, 12783 (2016).
  * (19) L. Zhu, X. Guo, C. Shu, H. Jeong, and S. Du, “Bright narrowband biphoton generation from a hot rubidium atomic vapor cell,” Appl. Phys. Lett. 110, 161101 (2017).
  * (20) J. Mika and L. Slodika, “High nonclassical correlations of large-bandwidth photon pairs generated in warm atomic vapor,” J. Phys. B: At. Mol. Opt. Phys. 53, 145501 (2020).
  * (21) T. Jeong and H. S. Moon, “Temporal- and spectral-property measurements of narrowband photon pairs from warm double-$\Lambda$-type atomic ensemble,” Opt. Express 28, 3985-3994 (2020).
  * (22) C.-Y. Hsu, Y.-S. Wang, J.-M. Chen, F.-C. Huang, Y.-T. Ke, E. K. Huang, W. Hung, K.-L. Chao, S.-S. Hsiao, Y.-H. Chen, C.-S. Chuu, Y.-C. Chen, Y.-F. Chen, I. A. Yu, “Generation of sub-MHz and spectrally-bright biphotons from hot atomic vapors with a phase mismatch-free scheme,” Opt. Express 29, 4632 (2021).
  * (23) K. Li, Y. Zhao,Y. Qin, Z. Chen, Y. Cai, and Y. Zhang, “Shaping Temporal Correlation of Biphotons in a Hot Atomic Ensemble,” Adv. Photonics Res. 2, 2100073 (2021).
  * (24) J.-M. Chen, C.-Y. Hsu, W.-K. Huang, S.-S. Hsiao, F.-C. Huang, Y.-H. Chen, C.-S. Chuu, Y.-C. Chen, Y.-F. Chen, and I. A. Yu, “Room-temperature biphoton source with a spectral brightness near the ultimate limit,” arXiv:2109.09062 (2021).
  * (25) C. Wang, C.-H. Lee, and Y.-H. Kim, “Generation and characterization of position-momentum entangled photon pairs in a hot atomic gas cell,” Opt. Express 27, 34611-34617 (2019).
  * (26) O. Davidson, R. Finkelstein, E. Poem, and O. Firstenberg, “Bright multiplexed source of indistinguishable single photons with tunable GHz-bandwidth at room temperature,” New J. Phys. 23, 073050 (2021).
  * (27) H. Jeong, H. Kim, J. Bae, J. Park, and H. S. Moon, “Doppler-broadened four-wave mixing under double-resonance optical pumping in the $5S_{1/2}-5P_{3/2}-4D_{5/2}$ transition of warm 87Rb atoms,” Opt. Express 29, 42384-42393 (2021).
  * (28) J. Park and H. S. Moon, “Generation of a bright four-photon entangled state from a warm atomic ensemble via inherent polarization entanglement,” Appl. Phys. Lett. 120, 024001 (2022).
  * (29) X. Guo, C. L. Zou, C. Schuck, H. Jung, R. Cheng, and H. X. Tang, “Parametric down-conversion photon-pair source on a nanophotonic chip,” Light: Sci. Appl. 6, e16249 (2017).
  * (30) C. Ma, X. Wang, V. Anant, A. D. Beyer, M. D. Shaw, and S. Mookherjea, “Silicon photonic entangled photon-pair and heralded single photon generation with CAR $>$ 12,000 and $g^{(2)}(0)<$ 0006,” Opt. Express 25, 32995-33006 (2017).
  * (31) L. Sansoni, K. H. Luo, C. Eigner, R. Ricken, V. Quiring, H. Herrmann, and C. Silberhorn, “A two-channel, spectrally degenerate polarization entangled source on chip,” NPJ Quantum Inf. 3, 5 (2017).
  * (32) R. R. Kumar, M. Raevskaia, V. Pogoretskii, Y. Jiao, and H. K. Tsang, “Entangled photon pair generation from an InP membrane micro-ring resonator,” Appl. Phys. Lett. 114, 021104 (2019).
  * (33) F. Samara, A. Martin, C. Autebert, M. Karpov, T. J. Kippenberg, H. Zbinden, and R. Thew, “High-rate photon pairs and sequential Time-Bin entanglement with Si3N4 microring resonators,” Opt. Express 27, 19309-19318 (2019).
  * (34) J. Zhao, C. Ma, M. Rüsing, and S. Mookherjea, “High Quality Entangled Photon Pair Generation in Periodically Poled Thin-Film Lithium Niobate Waveguides,” Phys. Rev. Lett. 124, 163603 (2020).
  * (35) G. Moody, L. Chang, T. J. Steiner, and J. E. Bowers, “Chip-scale nonlinear photonics for quantum light generation,” AVS Quantum Sci. 2, 041702 (2020).
  * (36) Y. Wang and K. D. Jöns and Z. Sun, “Integrated photon-pair sources with nonlinear optics,” Appl. Phys. Rev. 8, 011314 (2021).
  * (37) T. J. Steiner, J. E. Castro, L. Chang, Q. Dang,W. Xie, J. Norman, J. E. Bowers, and G. Moody, “Ultrabright Entangled-Photon-Pair Generation from an AlGaAs-On-Insulator Microring Resonator,” PRX Quantum 2, 010337 (2021).
  * (38) Y.-H. Chen, M.-J. Lee, I.-C. Wang, S. Du, Y.-F. Chen, Y.-C. Chen, and I. A. Yu, “Coherent Optical Memory with High Storage Efficiency and Large Fractional Delay,,” Phys. Rev. Lett. 110, 083601 (2013).
  * (39) D. Schraft, M. Hain, N. Lorenz, and T. Halfmann, “Stopped Light at High Storage Efficiency in a Pr3+:Y2SiO5 Crystal,” Phys. Rev. Lett. 116, 073602 (2016).
  * (40) Y.-W. Cho, T. Campbell, J. L. Evertt, J. Bernu, D. B. Higginbottom, M. T. Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, “Highly efficient optical quantum memory with long coherence time in cold atoms,” Optica 3, 100-107 (2016).
  * (41) P. Vernaz-Gris, K. Huang, M. Cao, A. S. Sheremet, and J. Laurat, “Highly-efficient quantum memory for polarization qubits in a spatially-multiplexed cold atomic ensemble,” Nat. Commun. 9, 363 (2018).
  * (42) Y.-F. Hsiao, P.-J. Tsai, H.-S. Chen, S.-X. Lin, C.-C. Hung, C.-H. Lee, Y.-H. Chen, Y.-F. Chen, I. A. Yu, and Y.-C. Chen, “Highly Efficient Coherent Optical Memory Based on Electromagnetically Induced Transparency,” Phys. Rev. Lett. 120, 183602 (2018).
  * (43) Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photon. 13, 346-351 (2019).
  * (44) P.-J. Tsai, Y.-F. Hsiao, and Y.-C. Chen, “Quantum storage and manipulation of heralded single photons in atomic memories based on electromagnetically induced transparency,” Phys. Rev. Research 2, 033155 (2020).
  * (45) A. Feizpour, M. Hallaji, G. Dmochowski, and A. M. Steinberg, “Observation of the nonlinear phase shift due to single post-selected photons,” Nat. Phys. 11, 905-909 (2015).
  * (46) K. M. Beck, M. Hosseini, Y. Duan, and V. Vuletić, “Large conditional single-photon cross-phase modulation,” Proc. Natl. Acad. Sci. U. S. A. 113, 9740-9744 (2016).
  * (47) D. Tiarks, S. Schmidt, G. Rempe, and S. Dürr, “Optical $\pi$ phase shift created with a single-photon pulse,” Science Adv. 2, e1600036 (2016).
  * (48) B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature (London) 536, 193-196 (2016).
  * (49) Z.-Y. Liu, Y.-H. Chen, Y.-C. Chen, H.-Y. Lo, P.-J. Tsai, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Large Cross-Phase Modulations at the Few-Photon Level,” Phys. Rev. Lett. 117, 203601 (2016).
  * (50) D. Tiarks, S. Schmidt-Eberle, T. Stolz, G. Rempe, and S. Dürr “A photon–photon quantum gate based on Rydberg interactions,” Nat. Phys. 15, 124-126 (2019).
  * (51) J. Vaneecloo, S. Garcia, and A. Ourjoumtsev, “Intracavity Rydberg Superatom for Optical Quantum Engineering: Coherent Control, Single-Shot Detection, and Optical $\pi$ Phase Shift,” Phys. Rev. X 12, 021034 (2022).
  * (52) T. Stolz, H. Hegels, M. Winter, B. Röhr, Y.-F. Hsiao, L. Husel, G. Rempe, and S. Dürr, “Quantum-Logic Gate between Two Optical Photons with an Average Efficiency above 40%,” Phys. Rev. X 12, 021035 (2022).
  * (53) A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich and T. A. B. Kennedy, “A quantum memory with telecom-wavelength conversion,” Nat. Phys. 6, 894-899 (2010).
  * (54) K.-F. Chang, T.-P. Wang, C.-Y. Chen, Y.-H. Chen, Y.-S. Wang, Y.-F. Chen, Y.-C. Chen, and I. A. Yu, “Low-loss high-fidelity frequency beam splitter with tunable split ratio based on electromagnetically induced transparency,” Phys. Rev. Research 3, 013096 (2021).
  * (55) C.-Y. Cheng, J.-J. Lee, Z.-Y. Liu, J.-S. Shiu, and Y.-F. Chen, “Quantum frequency conversion based on resonant four-wave mixing,” Phys. Rev. A 103, 023711 (2021).
  * (56) C.-Y Cheng, Z.-Y. Liu, P.-S. Hu, T.-N. Wang, C.-Y. Chien, J.-S. Shiu, I. A. Yu, Y.-C. Chen, and Y.-F. Chen, “Efficient frequency conversion based on electromagnetically induced transparency,” Opt. Lett. 46, 681-684 (2021).
  * (57) L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature 414, 413-418 (2001).
  * (58) N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145-195 (2002).
  * (59) R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photon. 3, 696-705 (2009).
  * (60) N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, “Quantum repeaters based on atomic ensembles and linear optics,” Rev. Mod. Phys. 83, 33-80 (2011).
  * (61) Q.-C. Sun, Y.-F. Jiang, Y.-L. Mao, L.-X. You, W. Zhang, W.-J. Zhang, X. Jiang, T.-Y. Chen, H. Li, Y.-D. Huang, X.-F. Chen, Z. Wang, J. Fan, Q. Zhang, and J.-W. Pan, “Entanglement swapping over 100 km optical fiber with independent entangled photon-pair sources,” Optica 4, 1214-1218 (2017).
  * (62) M. Riebe, T. Monzi, K. Kim, A. S. Villar, P. Schindler, M. Chwalla, M. Hennrich, and R. Blatt, “Deterministic entanglement swapping with an ion-trap quantum computer,” Nat. Phys. 4, 839-842 (2008).
  * (63) G. Andersson, M. K. Ekström, and P. Delsing, “Electromagnetically Induced Acoustic Transparency with a Superconducting Circuit,” Phys. Rev. Lett. 124, 240402 (2020).
  * (64) B.-m. Ann and G. A. Steele, “Tunable and weakly invasive probing of a superconducting resonator based on electromagnetically induced transparency,” Phys. Rev. A 102, 053721 (2020).
  * (65) A. M. Vadiraj, A. Ask, T. G. McConkey, I. Nsanzineza, C. W. S. Chang, A. F. Kockum, and C. M. Wilson, “Engineering the level structure of a giant artificial atom in waveguide quantum electrodynamics,” Phys. Rev. A 103, 023710 (2021).
  * (66) T. Northup, R. Blatt, “Quantum information transfer using photons,” Nat. Photonics 8, 356-363 (2014).
  * (67) D. D. Awschalom, R. Hanson, J. Wrachtrup, B. B. Zhou, “Quantum technologies with optically interfaced solid-state spins,” Nat. Photonics 12, 516-527 (2018).
  * (68) S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: A vision for the road ahead,” Science 362, eaam9288 (2018).
  * (69) Y.-H. Chen, M.-J. Lee, W. Hung, Y.-C. Chen, Y.-F. Chen, and I. A. Yu, “Demonstration of the Interaction between Two Stopped Light Pulses,” Phys. Rev. Lett. 108, 173603 (2012).
  * (70) B. Kim, K.-T. Chen, C.-Y. Hsu, S.-S. Hsiao, Y.-C. Tseng, C.-Y. Lee, S.-L. Liang, Y.-H. Lai, J. Ruseckas, G. Juzeliūnas, and I. A. Yu, “Effect of laser frequency fluctuation on the decay rate of Rydberg coherence,” Phys. Rev. A 100, 013815 (2019).
  * (71) S. Du, J. Wen, and M. H. Rubin, “Narrowband biphoton generation near atomic resonance,” J. Opt. Soc. Am. B 25, C98-C108 (2008).
  * (72) S. E. Harris and Y. Yamamoto, “Photon Switching by Quantum Interference,” Phys. Rev. Lett. 81, 3611 (1998).
  * (73) C.-Y. Wang, Y.-F. Chen, S.-C. Lin, W.-H. Lin, P.-C. Kuan, and I. A. Yu, “Low-light-level all-optical switching,” Opt. Lett. 31, 2350 (2006).
  * (74) Y.-H. Chen, M.-J. Lee, I.-C. Wang, and I. A. Yu, “Fidelity of electromagnetically-induced-transparency-based optical memory,” Phys. Rev. A 88, 023805 (2013).
  * (75) B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, “Narrow Band Source of Transform-Limited Photon Pairs via Four-Wave Mixing in a Cold Atomic Ensemble,” Phys. Rev. Lett. 111, 123602 (2013).
  * (76) Y.-F. Chen, S.-H. Wang, C.-Y. Wang, and I. A. Yu, “Manipulating the retrieved width of stored light pulses,” Phys. Rev. A 72, 053803 (2005).


# Rogue waves in multi-pair plasma medium

S. Khondaker∗, A. Mannan, N. A. Chowdhury, and A. A. Mamun  Department of
Physics, Jahangirnagar University, Savar, Dhaka-1342, Bangladesh
Email<EMAIL_ADDRESS>
###### Abstract

The nonlinear propagation of ion-acoustic (IA) waves (IAWs), which are
governed by the nonlinear Schrödinger equation (NLSE), in multi-pair plasmas
(MPPs) containing adiabatic positive and negative ion fluids as well as non-
extensive ($q$-distributed) electrons and positrons, is theoretically
investigated. It is observed that the MPP under consideration supports two
types of modes (namely, fast and slow IA modes), and the modulationally stable
and unstable parametric regimes for the fast and slow IA modes are determined
by the sign of the ratio of the dispersive coefficient to the nonlinear one.
It is also found that the modulationally unstable regime generates highly
energetic IA rogue waves (IARWs), and the amplitude as well as the width of
the IARWs decrease with increase in the value of $q$ (for both $q>0$ and $q<0$
limits). These new striking features of the IARWs are found to be applicable
in the space [viz. D-region ($\rm H^{+},O_{2}^{-}$) and F-region ($\rm
H^{+},H^{-}$) of the Earth’s ionosphere] and laboratory MPPs [viz. fullerene
($\rm C^{+},C^{-}$)].

###### keywords:

NLSE , modulational instability , rogue waves.

††journal: “High Energy Density Physics”

##  1 Introduction

Recently, the painstaking observational results support the existence of the
multi-pair plasmas (MPPs) not only in space environments (viz. cometary comae
[1], chromosphere, upper regions of Titan’s atmosphere [2], solar wind,
D-region ($\rm H^{+},O_{2}^{-}$) and F-region ($\rm H^{+},H^{-}$) of the
Earth’s ionosphere [3], etc.) but also laboratory situations (viz. Fullerene
($\rm C^{+},C^{-}$) [4], plasma processing reactors [5], neutral beam sources
[6], etc.). Plasma physicists are solely practicing wave dynamics, basically,
ion-acoustic (IA) waves (IAWs) in which restoring force is provided by the
electrons thermal pressure and moment of inertia is provided by the pair-ion
(PI) mass density. The adiabatic presure/dissipative force from the inertial
component of the plasma system provides a great effect to generate various
nonlinear electrostatic structures (viz. shocks, solitons, and envelope
solitons) in MPPs. The existence of positive ions in MPPs drastically alters
charge neutrality condition [7], the dispersive property [2], and nonlinearity
of the IAWs [7].

Boltzmann-Gibbs (BG) statistics, which was successful in describing thermally
equilibrium system, did not describe many natural and artificial complex
systems that are far from equilibrium. Motivated by this situation, Tsalli
came up with a concept of generalized BG statistics known as non-extensive
statistical mechanics that has offered a theoretical basis for interpreting
and analyzing non-equilibrium complex systems [8]. In non-extensive
$q$-distribution, the non-extensivity ($q$) plays an indispensable role in
executing the long range inter particle forces which includes Newtonian
gravitational forces and coulomb electric forces among charged particles in a
plasma medium. Many researchers have given more attention on $q$-distribution
to study the nonlinear property of the plasma medium [9, 10, 11]. Jannat et
al. [9] studied Gardner Solitons (GSs) in a non-extensive PI plasma (PIP).
Tribeche et al. [10] reported IA solitary structures in two component plasma
in presence of $q$-distributed electrons, and observed that the amplitude of
the potential pulse increases and more spiky with an increase in $q$. Ghosh et
al. [12] studied the effects of non-extensivity on the GSs in a non-extensive
electrons and positrons plasma medium. Eslami et al. [13] investigated the
stability of the IAWs in three component plasmas with $q$-distributed
electrons and positrons.

The field of modulational instability (MI) and rogue waves (RWs) has
considered one of the most interdisciplinary areas of research enclosing
plasma physics [7], oceanography [14], Bose-Einstein condensation [15], optics
[16], superfluid helium [17], and even finance [18] . Researchers have devoted
their attention to solve the mystery of this colossal waves due to the
intrinsically arbitrary nature, and the complex formation mechanisms.
Recently, a number of authors have studied different criteria of MI and RWs in
space and laboratory plasmas. Bains et al. [11] studied the MI of IAWs with a
$q$-distributed electrons, and found that the stable domain for IAWs increases
with $q$ within sub-extensive limit. Abdelwahed et al. [19] studied IA RWs
(IARWs) in PIP in presence of super-thermal electrons. Elwakil et al. [3]
analyzed the effects of non-thermal electrons on MI conditions of IAWs, and
observed that electrons non-thermality reduces the critical wave number
($k_{c}$) in PIP. El-Labany et al. [2] have derived the nonlinear Schrödinger
equation (NLSE) in a three component PIP in presence of iso-thermal electrons,
and observed that the number density of the negative ions enhances the
amplitude of the IARWs. In our present investigation, we have extended the
previous work of El-Labany et al. [2] by considering adiabatic pressure of the
PI in four component (comprising $q$-distributed electrons and positrons,
adiabatic positive and negative ions) PIP, and also will examine the effects
of the plasma parameters on the MI of IAWs, and the formation of the IARWS.

The outline of the paper is as follows: The governing equations describing our
plasma model are presented in Section 2. MI and RWs are given in Section 3. A
brief discussion is finally provided in Section 4.

##  2 Governing Equations

We consider an unmagnetized, one-dimensional collisionless four component MPPs
medium comprising of inertialess $q$-distributed electrons (mass $m_{e}$;
charge $q_{e}=-e$) and positrons (mass $m_{p}$; charge $q_{p}=+e$), inertial
adiabatic negative ions (mass $m_{-}$; charge $q_{-}=-eZ_{-}$) as well as
positive ions (mass $m_{+}$; charge $q_{+}=+eZ_{+}$), respectively. Here,
$Z_{-}$ ($Z_{+}$) is the charge state of negative (positive) ions. At
equilibrium, the charge neutrality yields the condition
$Z_{+}n_{+0}+n_{p0}=Z_{-}n_{-0}+n_{e0}$; where $n_{+0}$, $n_{-0}$, $n_{e0}$,
and $n_{p0}$, are the equilibrium number densities of positive ions, negative
ions, and $q$-distributed electrons and positrons, respectively. Now, the
basic set of equations in the normalized form can be written as

$\displaystyle\frac{\partial n_{-}}{\partial t}+\frac{\partial}{\partial
x}(n_{-}u_{-})=0,$ (1) $\displaystyle\frac{\partial u_{-}}{\partial
t}+u_{-}\frac{\partial u_{-}}{\partial x}+3\sigma_{1}n_{-}\frac{\partial
n_{-}}{\partial x}=\frac{\partial\phi}{\partial x},$ (2)
$\displaystyle\frac{\partial n_{+}}{\partial t}+\frac{\partial}{\partial
x}(n_{+}u_{+})=0,$ (3) $\displaystyle\frac{\partial u_{+}}{\partial
t}+u_{+}\frac{\partial u_{+}}{\partial x}+3\sigma_{2}n_{+}\frac{\partial
n_{+}}{\partial x}=-\mu_{1}\frac{\partial\phi}{\partial x},$ (4)
$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}=(\mu_{2}+\mu_{p}-1)n_{e}-\mu_{p}n_{p}+n_{-}-\mu_{2}n_{+}.$ (5)

(a)

(b)

Figure 1: The variation of $P/Q$ with $k$ at different values of $\mu_{1}$ for
(a) fast mode; (b) slow mode; along with fixed values of $\delta=1.3$,
$\mu_{2}=1.2$, $\mu_{p}=0.3$, $\sigma_{1}=0.005$, $\sigma_{2}=0.04$, and
$q=1.5$.

Here, the normalizing and associated parameters are defined as:
$n_{-}=N_{-}/n_{-0}$, $n_{+}=N_{+}/n_{+0}$, $n_{e}=N_{e}/n_{e0}$,
$n_{p}=N_{p}/n_{p0}$, $u_{-}=U_{-}/C_{-}$, $u_{+}=U_{+}/C_{-}$,
$\phi=e\tilde{\Phi}/k_{B}T_{e}$, $x=X/\lambda_{D_{-}}$, $t=T\omega_{P_{-}}$,
$C_{-}=\sqrt{(Z_{-}k_{B}T_{e}/m_{-})}$, $\lambda_{D_{-}}=\sqrt{k_{B}T_{e}/4\pi
e^{2}Z_{-}n_{-0}}$, $\omega_{P_{-}}=\sqrt{(4\pi
e^{2}Z_{-}^{2}n_{-0}/{m_{-}})}$,
$P_{-}=P_{-0}\left(N_{-}/n_{-0}\right)^{\gamma}$,
$P_{+}=P_{+0}\left(N_{+}/n_{+0}\right)^{\gamma}$, $P_{-0}=n_{-0}k_{B}T_{-}$,
$P_{+0}=n_{+0}k_{B}T_{+}$, $\gamma=(N+2)/N$, $\mu_{1}=Z_{+}m_{-}/Z_{-}m_{+}$,
$\mu_{2}=Z_{+}n_{+0}/Z_{-}n_{-0}$, $\mu_{p}=n_{p0}/Z_{-}n_{-0}$,
$\sigma_{1}=T_{-}/Z_{-}T_{e}$, $\sigma_{2}=T_{+}m_{-}/Z_{-}T_{e}m_{+}$; where
$n_{-}$, $n_{+}$, $u_{-}$, $u_{+}$, $x$, $t$, $\phi$, $C_{-}$, $\lambda_{D-}$,
$\omega_{P-}$, $k_{B}$ $T_{-}$, $T_{+}$, $T_{e}$, $T_{p}$, $P_{-0}$, and
$P_{+0}$ is the number densities of negative ion, positive ion, electron,
positron, negative ions fluid speed, positive ions fluid speed, space co-
ordinate, time co-ordinate, electro-static potential, sound speed of the
negative ions, Debye length of the negative ions, angular frequency of the
negative ions, Boltzmann constant, negative ions temperature, positive ions
temperature, $q$-distributed electron temperature, $q$-distributed positron
temperature, the equilibrium adiabatic pressure of negative ions, and the
equilibrium adiabatic pressure of positive ions, respectively. $N$ is the
degree of freedom chosen to be $1$ for one-dimensional adiabatic case, hence
$\gamma$=3. It may be pointed here that the mass of the positive ion is
greater than the negative ion, i.e., $m_{+}>m_{-}$, the number density of the
positive ions is greater than the negative ions, i.e., $n_{+0}>n_{-0}$, and
electron and positron temperatures are greater than the negative and positive
ion temperature, i.e., $T_{e}$, $T_{p}>T_{-}$, $T_{+}$. Now, the expressions
for the normalized electron and positron number density obeying
$q$-distribution is given by [7]

$\displaystyle
n_{e}=\left[1+(q-1)\phi\right]^{\frac{(q+1)}{2(q-1)}}=1+n_{1}\phi+n_{2}\phi^{2}+n_{3}\phi^{3}+\cdot\cdot\cdot,$
(6) $\displaystyle
n_{p}=\left[1-(q-1)\delta\phi\right]^{\frac{(q+1)}{2(q-1)}}=1-n_{1}\delta\phi+n_{2}\delta^{2}\phi^{2}-n_{3}\delta^{3}\phi^{3}+\cdot\cdot\cdot,$
(7)

where

$\displaystyle n_{1}=(1+q)/2,~{}~{}~{}n_{2}=-(1+q)(q-3)/8,$ $\displaystyle
n_{3}=(1+q)(q-3)(3q-5)/48,~{}~{}~{}\delta=T_{e}/T_{p},~{}~{}~{}\mbox{and}~{}~{}~{}T_{e}>T_{p}.\
$

Here, the parameter $q$, generally known as entropic index, quantifies the
degree of non-extensivity. It is noteworthy that when $q=1$, the entropy
reduces to standard Maxwell-Boltzmann distribution. On the other hand, in the
limits $q>1$ ($q<1$), the entropy shows sub-extensivity (super-extensivity).
Now, by substituting (7) into (5) and expanding up to third order, we get

$\displaystyle\frac{\partial^{2}\phi}{\partial
x^{2}}=(\mu_{2}-1)+n_{-}-\mu_{2}n_{+}+\gamma_{1}\phi+\gamma_{2}\phi^{2}+\gamma_{3}\phi^{3}\cdot\cdot\cdot,$
(8)

where

$\displaystyle\gamma_{1}=n_{1}(\mu_{2}+\mu_{p}-1+\mu_{p}\delta),$
$\displaystyle\gamma_{2}=n_{2}(\mu_{2}+\mu_{p}-1-\mu_{p}\delta^{2}),$
$\displaystyle\gamma_{3}=n_{3}(\mu_{2}+\mu_{p}-1+\mu_{p}\delta^{3}).\ $

To analyze the MI of the IAWs, we will derive the NLSE by employing the
reductive perturbation method. So, the independent stretched variables as
$\xi=\epsilon(x-v_{g}t)$ and $\tau=\epsilon^{2}t$, where $\epsilon$ is a small
perturbation parameter and $v_{g}$ is the envelope group velocity.
Furthermore, the dependent variables can be expanded as

$\displaystyle\Lambda(x,t)=\Lambda_{0}+\sum_{m=1}^{\infty}\epsilon^{(m)}\sum_{l=-\infty}^{\infty}\Lambda_{l}^{m}(\xi,\tau)\exp(il\Upsilon)$
(9)

where $\Lambda_{l}^{m}$ =
$[n_{-l}^{(m)},u_{-l}^{(m)},n_{+l}^{(m)},u_{+l}^{(m)},\phi_{l}^{(m)}]^{T}$,
$\Lambda_{0}$ = $[1,0,1,0,0]^{T}$, $\Upsilon=(kx-\omega t)$, and $\omega$
($k$) represents the angular frequency (wave number) of the carrier waves,
respectively. We are going to parallel mathematical steps as Chowdhury et al.
[20] have done in their work to find successively the IAWs dispersion
relation, group velocity, and NLSE. The IAWs dispersion relation

$\displaystyle\omega^{2}=\frac{M\pm k^{2}\sqrt{M^{2}-4GN}}{2G},$ (10)

where $G=(\gamma_{1}+k^{2})$,
$M=(1+\mu_{1}\mu_{2}+\gamma_{1}\lambda_{1}+\gamma_{1}\lambda_{2}+\lambda_{1}k^{2}+\lambda_{2}k^{2})$,
$N=(\lambda_{2}+\mu_{1}\mu_{2}\lambda_{1}+\gamma_{1}\lambda_{1}\lambda_{2}+\lambda_{1}\lambda_{2}k^{2})$,
$\lambda_{1}=3\sigma_{1}$, and $\lambda_{2}=3\sigma_{2}$. In order to get real
and positive value of $\omega$, the condition $M^{2}>4GN$ is required to
maintain. It is clear from (10), there exist two distinct modes that depend on
the signs $\pm$. The positive sign refers to fast ($\omega_{f}$) IA mode
whereas negative sign refers to slow ($\omega_{s}$) IA mode. The group
velocity of the IAWs is given by

$\displaystyle v_{g}=\frac{\partial\omega}{\partial
k}=\frac{F_{1}-2A^{2}S^{2}-AS(A-\mu_{1}\mu_{2}S)}{2\omega
k(A^{2}+\mu_{1}\mu_{2}S^{2})},$ (11)

Where
$F_{1}=\omega^{2}(A^{2}+\mu_{1}\mu_{2}S^{2})+k^{2}(\lambda_{1}A^{2}+\mu_{1}\mu_{2}\lambda_{2}S^{2})$,
$S=\lambda_{1}k^{2}-\omega^{2}$, and $A=\omega^{2}-\lambda_{2}k^{2}$. Finally,
we obtain the standard NLSE:

$\displaystyle
i\frac{\partial\Phi}{\partial\tau}+P\frac{\partial^{2}\Phi}{\partial\xi^{2}}+Q|\Phi|^{2}\Phi=0,$
(12)

where $\Phi=\phi_{1}^{(1)}$ for simplicity. The dispersion coefficient $P$ and
the nonlinear coefficient $Q$ are, respectively, given by

$\displaystyle P=\frac{F_{2}-A^{3}S^{3}}{2AS\omega
k^{2}(A^{2}+\mu_{1}\mu_{2}S^{2})},$ (13) $\displaystyle
Q=\frac{A^{2}S^{2}\\{2\gamma_{2}(C_{5}+C_{10})+3\gamma_{3}-F_{3}\\}}{2\omega
k^{2}(A^{2}+\mu_{1}\mu_{2}S^{2})},$ (14)

where

$\displaystyle F_{2}=A^{3}\\{(\omega
v_{g}-\lambda_{1}k)(\lambda_{1}k^{3}+k\omega^{2}-2\omega v_{g}k^{2}-kS)$
$\displaystyle+(v_{g}k-\omega)(\omega^{3}+\lambda_{1}\omega
k^{2}-2v_{g}k\omega^{2}-v_{g}kS)\\}$
$\displaystyle-\mu_{1}\mu_{2}S^{3}\\{(\omega
v_{g}-\lambda_{2}k)(\lambda_{2}k^{3}+k\omega^{2}-2\omega v_{g}k^{2}+kA)$
$\displaystyle+(v_{g}k-\omega)(\omega^{3}+\lambda_{2}\omega
k^{2}-2v_{g}k\omega^{2}+kv_{g}A)\\},$ $\displaystyle F_{3}=\frac{2\omega
k^{3}(C_{2}+C_{7})}{S^{2}}+\frac{(\omega^{2}k^{2}+\lambda_{1}k^{4})(C_{1}+C_{6})}{S^{2}}$
$\displaystyle+\frac{\mu_{1}\mu_{2}(\omega^{2}k^{2}+\lambda_{2}k^{4})(C_{3}+C_{8})}{A^{2}}$
$\displaystyle+\frac{2\mu_{1}\mu_{2}\omega k^{3}(C_{4}+C_{9})}{A^{2}},\ $
$\displaystyle
C_{1}=\frac{2C_{5}k^{2}S^{2}-3\omega^{2}k^{4}-\lambda_{1}k^{6}}{2S^{3}},$
$\displaystyle C_{2}=\frac{\omega(C_{1}S^{2}-k^{4})}{kS^{2}},$ $\displaystyle
C_{3}=\frac{2C_{5}\mu_{1}k^{2}A^{2}+\mu_{1}^{2}(\lambda_{2}k^{6}+3\omega^{2}k^{4})}{2A^{3}},$
$\displaystyle C_{4}=\frac{\omega(C_{3}A^{2}-\mu_{1}^{2}k^{4})}{kA^{2}},$
$\displaystyle
C_{5}=\frac{A^{3}(3\omega^{2}k^{4}+\lambda_{1}k^{6}-2\gamma_{2}S^{3})+\mu_{2}S^{3}(3\omega^{2}\mu_{1}^{2}k^{4}+\lambda_{2}\mu_{1}^{2}k^{6})}{2A^{3}S^{3}(4k^{2}+\gamma_{1})+2k^{2}S^{2}A^{3}-\mu_{1}\mu_{2}k^{2}A^{2}S^{3}},$
$\displaystyle C_{6}=\frac{k^{2}\omega^{2}+2v_{g}\omega
k^{3}+\lambda_{1}k^{4}-C_{10}S^{2}}{S^{2}(v_{g}^{2}-\lambda_{1})},$
$\displaystyle C_{7}=\frac{v_{g}C_{6}S^{2}-2\omega k^{3}}{S^{2}},$
$\displaystyle C_{8}=\frac{\mu_{1}^{2}(2\omega
v_{g}k^{3}+k^{2}\omega^{2}+\lambda_{2}k^{4})-\mu_{1}C_{10}A^{2}}{A^{2}(v_{g}^{2}-\lambda_{2})},$
$\displaystyle C_{9}=\frac{v_{g}C_{8}A^{2}-2\omega\mu_{1}^{2}k^{3}}{A^{2}},$
$\displaystyle
C_{10}=\frac{F_{4}+2\gamma_{2}A^{2}S^{2}(v_{g}^{2}-\lambda_{1})(v_{g}^{2}-\lambda_{2})}{F_{5}-\gamma_{1}A^{2}S^{2}(v_{g}^{2}-\lambda_{1})(v_{g}^{2}-\lambda_{2})},$
$\displaystyle F_{4}=A^{2}(k^{2}\omega^{2}+2v_{g}\omega
k^{3}+\lambda_{1}k^{4})(v_{g}^{2}-\lambda_{2})$
$\displaystyle-\mu_{2}S^{2}(\mu_{1}^{2}k^{2}\omega^{2}+2\omega
v_{g}\mu_{1}^{2}k^{3}+\lambda_{2}\mu_{1}^{2}k^{4})(v_{g}^{2}-\lambda_{1}),$
$\displaystyle
F_{5}=A^{2}S^{2}(v_{g}^{2}-\lambda_{2})+\mu_{1}\mu_{2}A^{2}S^{2}(v_{g}^{2}-\lambda_{1}).$

(a)

(b)

Figure 2:  The variation of $|\Phi|$ with $\xi$ at different values of (a)
$\mu_{1}$ and $\mu_{2}=1.2$; (b) $\mu_{2}$ and $\mu_{1}=0.5$; along with
$\delta=1.3$, $\mu_{p}=0.3$, $\sigma_{1}=0.005$, $\sigma_{2}=0.04$, $q=1.5$,
$k=0.35$, $\tau=0$, and $\omega_{f}$.

##  3 MI analysis and Rogue waves

The stability of IAWs in four component plasma medium is governed by the sign
of $P$ and $Q$ [21, 22, 23]. When $P$ and $Q$ are same sign ($P/Q>0$), the
evolution of the IAWs amplitude is modulationally unstable whereas when $P$
and $Q$ are opposite sign ($P/Q<0$), the IAWs are modulationally stable in
presence of the external perturbations. The plot of $P/Q$ against $k$ yields
stable and unstable domain for the IAWs. The point, at which transition of
$P/Q$ curve intersect with $k$-axis, is known as threshold or critical wave
number $k$ ($=k_{c}$). We have depicted $P/Q$ versus $k$ curve with $\mu_{1}$
for both fast and slow IA modes in Figs. 1a and 1b, respectively, and it is
obvious from these figures that (i) both stable and unstable domain can be
found for both fast and slow IA modes; (ii) the $k_{c}$ decreases (increases)
with increasing in the values of negative ion mass for fast (slow) IA mode
when $Z_{-}$, $Z_{+}$, and $m_{+}$ remain invariant. So, the mass of the
positive ion plays a vital role in recognizing the stabile domain of the IAWs.

(a)

(b)

Figure 3:  The variation of $|\Phi|$ with $\xi$ at different values of (a)
$\delta$ and $\mu_{p}=0.3$; (b) $\mu_{p}$ and $\delta=1.3$ ; along with
$\mu_{1}=0.5$, $\mu_{2}=1.2$, $\sigma_{1}=0.005$, $\sigma_{2}=0.04$, $q=1.5$,
$k=0.35$, $\tau=0$, and $\omega_{f}$.

The governing equation for the highly energetic IARWs in the unstable region
($P/Q>0$) can be written as [24, 25]

$\displaystyle\Phi(\xi,\tau)=\sqrt{\frac{2P}{Q}}\left[\frac{4(1+4iP\tau)}{1+16P^{2}\tau^{2}+4\xi^{2}}-1\right]\mbox{exp}(2iP\tau).$
(15)

We have numerically analyzed (15) in Figs. 2a and 2b to understand the effects
of the mass and the number density of the negaitve/positive ions, in fact the
charge state of negaitve/positive ions, on the formation of the IARWs, and it
is clear from these figures that (a) the nonlinearity of the DPP medium
decreases, by depicting smaller amplitude of the IARWs, with negative ion mass
for constant values of positive ion mass, charge state of the positive and
negative ions (via $\mu_{1}$); (b) the electrostatic IARWs potential increases
with the increase in the value of negative ion number density ($n_{-0}$), but
decreases with increase of the positive ion number density ($n_{+0}$) for
constant value of $Z_{+}$ and $Z_{-}$ (via $\mu_{2}$); (c) physically,
negative ions lead to enhance the nonlinearity of the plasma medium
successively the amplitude of the electrostatic IARWs potential. This result
agrees with the result of El-Labany et al. [2] work.

(a)

(b)

Figure 4: The variation of $|\Phi|$ with $\xi$ at different values of $q$ for
(a) $q=+$ve; (b) $q=-$ve; along with $\delta=1.3$, $\mu_{1}=0.5$,
$\mu_{2}=1.2$, $\mu_{p}=0.3$, $\sigma_{1}=0.005$, $\sigma_{2}=0.04$, $k=0.35$,
$\tau=0$, and $\omega_{f}$.

Figure 3a is illustrated to demonstrate the effects of electron-to-positron
temperature ratio on the IARWs. It clearly shows that the electrostatic IARWs
potential decreases with increasing electron-to-positron temperature ratio
(via $\delta$). Similarly, from Fig. 3b it can be shown that the electrostatic
IARWs potential decreases with the increase in the value of positron number
density ($n_{p0}$), but increases with increase of the negative ion number
density ($n_{-0}$) for constant value of $Z_{-}$ (via $\mu_{p}$).

We have also numerically analyzed (15) in Figs. 4a and 4b to illustrate the
influence of $q-$distributed electrons on the formation of IARWs and it is
obvious that (i) the amplitude of the IARWs decreases with $q$ (for both $q>0$
and $q<0$); (ii) the amplitude of electrostatic wave is independent of the
sign of $q$, but dependent on the magnitude of $q$, and this result is a good
agreement with Chowdhury et al. [7] work; (iii) these figures, in fact,
indicate a comparison between electrostatic potential for $q>0$ and $q<0$;
(iv) the amplitude of the IARWs electrostatic potential for same interval of
positive $q$ is not same as negative $q$.

##  4 Discussion

In this study, we have performed a nonlinear analysis of IAWs in a MPPs
consisting of inertialess $q$-distributed electrons and positrons, inertial
adiabatic negative as well as positive ions. The evolution of the IAWs is
governed by the standard NLSE and the nonlinear $P$ and the dispersive $Q$
coefficients represent the stable/unstable domain of the IAWs in presence of
the external perturbation. This theoretical investigation give rises to some
noteworthy results that can be summed up as follows:

  1. 1.

Both fast and slow IA modes admit stable and unstable domains for the IAWs.

  2. 2.

The nonlinearity of the MPPs decreases with negative ion mass for constant
values of positive ion mass, charge state of the positive and negative ions
(via $\mu_{1}$).

  3. 3.

The electrostatic IARWs potential increases (decreases) with an increase in
the value of negative ion number density (positive ion density) for invariant
charge state of the positive and negative ions (via $\mu_{2}$).

  4. 4.

IARWs potential decreases with increasing electron-to-positron temperature
ratio (via $\delta$).

  5. 5.

IARWs potential decreases (increases) with the increase in the value of
positron number density (negative ion number density) for constant charge
state of negative ion (via $\mu_{p}$).

  6. 6.

The amplitude and the width of the IARWs decrease with $q$ (for both $q>0$ and
$q<0$).

Finally, we hope that the results from our present theoretical investigation
may be helpful in understanding the MI of IAWs and generation of the IARWs in
laboratory plasmas, viz. PI fullerene ($\rm C^{+},C^{-}$) [4] or in the space
plasma, viz. D-region ($\rm H^{+},O_{2}^{-}$) and F-region ($\rm H^{+},H^{-}$)
of the Earth’s ionosphere [3], and Titan’s atmosphere [2].

## Acknowledgement

S. Khondaker is grateful to the Bangladesh Ministry of Science and Technology
for awarding the National Science and Technology (NST) Fellowship.

## References

  * [1] P. H. Chaizy, H. Rème, J. A. Sauvaud, C. D’uston, R. P. Lin, D. E. Larson, D. L. Mitchell, K. A. Anderson, C. W. Carlson, A. Korth, and D. A. Mendis, Nature (London) 349, 393 (1991).
  * [2] S.K. El-Labany, W.M. Moslem, N.A. El-Bedwehy, R. Sabry, H.N.A. El-Razek, Astrophys. Space Sci. 338 3 (2012).
  * [3] S.A. Elwakil, E.K. El-Shewy, H.G. Abdelwahed, Phys. Plasmas 17 052301 (2010).
  * [4] R. Sabry, Phys. Plasmas 15 092101 (2008).
  * [5] R.A. Gottscho, C.E. Gaebe, IEEE Trans. Plasma Sci. 14 92 (1986).
  * [6] M. Bacal, G.W. Hamilton, Phys. Rev. Lett. 42 1538 (1979).
  * [7] N. A. Chowdhury, A. Mannan, M. M. Hasan, and A. A. Mamun, Chaos 27, 093105 (2017).
  * [8] C. Tsallis, J. Stat. Phys. 52, 479 (1988).
  * [9] N. Jannat, M. Ferdousi, and A. A. Mamun, Plasma Phys. Rep. 42, 678 (2016).
  * [10] M. Tribeche, L. Djebarni, R. Amour, Phys. Plasmas 17, 042114 (2010).
  * [11] A. S. Bains, M. Tribeche, and T. S. Gill, Phys. Plasmas 18, 022108 (2011).
  * [12] U. N. Ghosh, D. K. Ghosh, P. Chatterjee, M. Bacha, M. Tribeche, Phys. Plasmas 343, 265 (2013).
  * [13] P. Eslami, M. Mottaghizadeh, and H. R. Pakzad, Phys. Plasmas 18, 102313 (2011).
  * [14] C. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue Waves in the Ocean (Springer-Verlag, Berlin, 2009).
  * [15] Yu. V. Bludov, V. V. Konotop, N . Akhmediev . Phys. Rev. A 80, 033610 (2009)
  * [16] D. R. Solli, C. Ropers, P. Koonath and B. Jalali, Nature, 450, 1054 (2007)
  * [17] A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, P. V. E. McClintock, Phys. Rev. Lett. 101 065303 (2008).
  * [18] Z. Yan, Commun. Theor. Phys. 54, 947 (2010).
  * [19] H.G. Abdelwahed, E.K. El-shewy, M.A. Zahran, S.A. Elwakil, Phys. Plasmas 23 (2016) 022102.
  * [20] N.A. Chowdhury, M.M. Hasan, A. Mannan, A.A. Manun, Vacuum 147 (2018) 31.
  * [21] S Sultana I. Kourakis, Plasma Phys. Control. Fusion 53 (2011) 045003.
  * [22] R. Fedele, Phys. Scr. 65 (2002) 502.
  * [23] I. Kourakis P. K. Shukla, Nonlinear Proc. Geophys. 12 (2005) 407.
  * [24] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Phys. Rev. E 80 (2009) 026601.
  * [25] A. Ankiewicz, N. Devine, N. Akhmediev, Phys. Lett. A 373 (2009) 3997.