# Annihilation and recurrence of vortex–antivortex pairs in two-component
Bose–Einstein condensates

Junsik Han1 Makoto Tsubota1,2,3 1Department of Physics, Osaka City University,
3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan  2The Advanced Research
Institute for Natural Science and Technology, Osaka City University, 3-3-138
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan  3Nambu Yoichiro Institute of
Theoretical and Experimental Physics (NITEP), Osaka City University, 3-3-138
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan

###### Abstract

The annihilation of vortex–antivortex pairs is a key event in two-dimensional
Bose–Einstein condensates (BECs). It is known that dissipation or a catalyst
vortex is required for the annihilation of the pairs in one-component BECs. We
numerically confirmed in two-component BECs that the pairs can be annihilated
even without any dissipation or catalyst vortices when the intercomponent
interaction is strong. In addition, the pair is recreated alternately in two
components under certain conditions, which we call the recurrence.

##  I Introduction

Quantum turbulence (QT) has been studied in superfluid helium for more than
half a century and in atomic Bose–Einstein condensates (BECs) during the past
decades Tsubota  _et al._ (2013); Barenghi  _et al._ (2014); Madeira  _et al._
(2020). QT consists of a large number of quantized vortices, which are well-
defined topological defects with quantized circulation. The core size of the
quantized vortex is characterized by the coherence length $\xi$, and the
diluteness of BEC gas makes $\xi$ on the order of $\mu\rm{m}$; consequently,
optical techniques make the core visible. In addition, the sign of the
circulation of vortices is confirmed by the Bragg scattering method Seo  _et
al._ (2017). Thus, QT is studied by focusing on the dynamics of the vortices
in BECs both theoretically and experimentally.

In BEC systems, two-dimensional (2D) QT has been studied. The 2D QT is
realized by using a tightly confining trap potential along one axis and
restricting the motion of the vortices in the 2D space. In 2D QT, the
formation of clusters of like-sign vortices has been studied extensively White
_et al._ (2012); Neely  _et al._ (2013); Billam  _et al._ (2014); Simula  _et
al._ (2014); Reeves  _et al._ (2014); Yu  _et al._ (2016); Groszek  _et al._
(2016); Salman and Maestrini (2016); Gauthier  _et al._ (2019); Johnstone  _et
al._ (2019). It is thought that this phenomenon is related to the inverse
energy cascade Kraichnan (1967); Reeves  _et al._ (2013), wherein the energy
is transferred from a small to a large spatial scale and the vortices create a
large structure.

In 2D systems, the dynamics of a vortex–antivortex pair is one of the simplest
and most important elements of the dynamics of the vortices. If the system has
no dissipation, a pair is stable and moves with a self-induced constant
velocity. However, a vortex pair can be annihilated through three mechanisms.
First is through dissipation. When the fluid is incompressible, a dissipative
mechanism can annihilate a vortex pair Kim  _et al._ (2016). Even when the
fluid is compressible, a vortex pair maintains the motion without dissipation.
If some dissipation is included, a vortex pair gradually decreases the
distance between two vortices and eventually gets annihilated when emitting
the density waves. Second is through a many-body process. When catalyst
vortices interact with a pair, they can annihilate the pair. This process has
been studied in the decay of QT Nazarenko and Onorato (2007); Kwon  _et al._
(2014); Stagg  _et al._ (2015); Cidrim  _et al._ (2016); Groszek  _et al._
(2016); Karl and Gasenzer (2017); Baggaley and Barenghi (2018); Groszek  _et
al._ (2020). Four-body (two catalyst vortices) Cidrim  _et al._ (2016);
Groszek  _et al._ (2016); Baggaley and Barenghi (2018); Groszek  _et al._
(2020), three-body (one catalyst vortex) Nazarenko and Onorato (2007); Groszek
_et al._ (2016); Baggaley and Barenghi (2018); Groszek  _et al._ (2020), two-
body (no catalyst vortex) Kwon  _et al._ (2014); Stagg  _et al._ (2015);
Cidrim  _et al._ (2016); Groszek  _et al._ (2016); Baggaley and Barenghi
(2018); Groszek  _et al._ (2020), and one-body (drift out from the trap) Kwon
_et al._ (2014); Stagg  _et al._ (2015); Cidrim  _et al._ (2016); Groszek  _et
al._ (2016, 2020) processes have been studied with and without dissipation.
Third is through effective dissipation due to phonons Shukla  _et al._ (2014),
however, this process can be negligible in our case one .

In this paper, we present another novel scenario of vortex pair annihilation
in two-component BECs. Multicomponent BECs have also been studied in two
dimensions Kasamatsu  _et al._ (2003, 2005); Eto  _et al._ (2011); Nakamura
_et al._ (2012); Kasamatsu  _et al._ (2016); Karl  _et al._ (2013) and three
dimensions Takeuchi  _et al._ (2010); Ishino  _et al._ (2011). In two-
component BECs without dissipation, the total energy of the system is
conserved; however, they exchange energies through the intercomponent
interaction. Then, we found that a vortex pair can be annihilated even without
dissipation or catalyst vortices when the repulsive intercomponent interaction
is sufficiently strong. Here, the repulsive intercomponent interaction causes
the exchange between the energy of each component and the intercomponent-
interaction energy. This energy exchange results in drastic dynamics, namely,
the recurrence. After a vortex pair is annihilated in one component, another
vortex pair appears in another component. Thereafter, the pair gets
annihilated and a vortex pair appears again in the original component. The two
components repeat the pair annihilation and creation by exchanging the kinetic
energy between them. Whether this recurrence occurs or whether the pair just
gets annihilated depends on the system size. In coherently coupled BECs, the
periodic vorticity transfer between the two components is also confirmed
Calderaro  _et al._ (2017). However, this transfer comes from Rabi coupling
depending on the phase difference between the two components, which is
different from the recurrence in this paper.

This recurrence accompanies the creation of rarefaction pulses, which are dark
blobs without vorticity Jones and Roberts (1982); Berloff (2002); Feijoo  _et
al._ (2017). Jones and Roberts studied the 2D Gross–Pitaevski equation with
axisymmetric disturbances, and found that the dispersion relation shows a
continuous sequence of a vortex pair and a rarefaction pulse Jones and Roberts
(1982). This study assumed a one-component uniform condensate, although the
scenario is also available in such systems as trapped or two-component BECs.
In general, a vortex can be nucleated from a low-density region, e.g., owing
to a repulsive potential Sasaki  _et al._ (2010); Fujimoto and Tsubota (2011)
or to the outskirt of the trapped condensate Tsubota  _et al._ (2002). Even in
a uniform system, a rarefaction pulse triggers a low-density region, from
which a vortex pair appears in a one-component system under some strong
disturbance Berloff (2002); Feijoo  _et al._ (2017). In two-component BECs,
the intercomponent interaction enables the continuous transformation between a
vortex pair and a rarefaction pulse; a similar phenomenon is observed
numerically in three-dimensional two-component BECs Takeuchi  _et al._ (2010).
Thus, a rarefaction pulse is also created alternately in both components
whenever a vortex pair is created through the recurrence.

The rest of the paper is organized as follows. In section II, we introduce the
numerical model and explain the assumption of the system. In section III, we
show the two results of our simulation. The pair annihilation without
dissipation is presented in section III.1. In section III.2, we describe the
creation of the rarefaction pulse and the recurrence of a vortex pair. We
discuss the distribution of the kinetic energy when a rarefaction pulse or a
vortex pair is created in section III.3. Finally, section IV provides the
conclusions.

##  II Model

In this paper, we focus on 2D two-component BECs. The Gross–Pitaevskii(GP)
equation can quantitatively describe many experimental results Pethick and
Smith (2008); Tsatsos  _et al._ (2016); Tsubota  _et al._ (2017). The GP
equations in two-component uniform BECs are written as

$\begin{array}[]{l}\mathrm{i}\hbar\frac{\partial}{\partial
t}\psi_{j}({\bm{r}},t)\\\ \\\
=\left[-\frac{\hbar^{2}}{2m_{j}}\nabla^{2}+\displaystyle\sum_{j^{\prime}=1,2}g_{jj^{\prime}}|\psi_{j^{\prime}}({\bm{r}},t)|^{2}-\mu_{j}\right]\psi_{j}({\bm{r}},t),\\\
\\\ \hskip 12.0pt(j=1,2)\end{array}$ (1)

In these equations,
$\psi_{j}=\sqrt{n_{j}({\bm{r}},t)}\mathrm{e}^{\mathrm{i}\theta_{j}({\bm{r}},t)}$
is the macroscopic wave function of the $j$-th component, where
$n_{j}({\bm{r}},t)$ is the density of the condensate and
$\theta_{j}({\bm{r}},t)$ is its phase. Here, $m_{j}$ is the mass of the atom,
$g_{jj^{\prime}}$ is the intensity of atom–atom interaction, $\mu_{j}$ is the
chemical potential, $\hbar=h/(2\pi)$, and $h$ is Planck’s constant. For
simplicity, we choose $m_{1}=m_{2}=m$ and $g_{11}=g_{22}=g$. Throughout the
paper, we use Eq. (1) without any dissipation term.

In this paper, we consider the case in which component 1 contains a vortex
pair and component 2 has no pair in the initial state. We imprint vortices in
component 1 by multiplying the wave function by a phase factor
$\displaystyle\Pi_{i}\exp(\mathrm{i}\phi_{i})$, with
$\phi_{i}(x,y)=s_{i}\arctan[(y-y_{i})/(x-x_{i})]$. Here, the coordinates
$(x_{i},y_{i})$ and $s_{i}$ refer to the position and the sign of the
circulation of the $i$-th vortex ($i=1,2$), respectively. After the
imprinting, the wave function evolves in imaginary time. When the vortex cores
are formed and the distance between two vortices becomes $5.75\xi$, we stop
the imaginary time evolution and treat this state as the initial state of the
real time evolution Ruostekoski and Dutton (2005); Groszek  _et al._ (2016).
Then, stopping the imaginary time evolution leaves the residual energy due to
phonons, however, this hardly affects the dynamics studied in this paper.

In this calculation, Eq. (1) is made dimensionless by the length scale
$\xi=\hbar/\sqrt{mgn}$, with the time scale $\tau=\hbar/gn$, where $n$ is the
density of the uniform condensate. The dimensionless wave function is defined
as
$\tilde{\psi}_{j}({\bm{r}},t)=\displaystyle\frac{1}{\sqrt{n}}\psi_{j}({\bm{r}},t)$,
and the unit of energy is $gn$. We perform these numerical calculations under
the periodic boundary condition with three different periods of $L=64$, $32$,
and $16$. Here, the space is divided by an $N^{2}$ uniform Cartesian mesh, and
we consider the resolution of the space $dx=dy=L/N=0.25$ for $L=64,32$ and
$dx=dy=L/N=0.0625$ for $L=16$. We solve the dimensionless GP equation using
the Fourier spectrum method and the fourth-order Runge–Kutta method. In the
remaining paper, we use the dimensionless quantities.

It is important to divide the total energy of the system into components in
the study of quantum hydrodynamics Nore  _et al._ (1997); Kobayashi and
Tsubota (2005). In the present system, the energy $E_{j}(t)$ of the $j$-th
component is composed of the kinetic energy $E_{{\rm kin},j}(t)$, quantum
energy $E_{{\rm q},j}(t)$, and intracomponent-interaction energy $E_{{\rm
int},j}(t)$,

$E_{j}(t)=E_{{\rm kin},j}(t)+E_{{\rm q},j}(t)+E_{{\rm int},j}(t),$ (2)

These energies are defined as

$E_{{\rm kin},j}(t)=\frac{1}{2}\int\left[\sqrt{n_{j}({\bf
r},t)}\nabla\theta_{j}({\bf r},t)\right]^{2}d{\bf r},$ (3) $E_{{\rm
q},j}(t)=\frac{1}{2}\int[\nabla n_{j}({\bf r},t)]^{2}d{\bf r},$ (4) $E_{{\rm
int},j}(t)=\frac{g}{2}\int n_{j}({\bf r},t)^{2}d{\bf r},$ (5)

The kinetic energy is divided into the incompressible kinetic energy $E_{{\rm
kin},j}^{\rm i}(t)$, owing to the quantized vortices, and the compressible
kinetic energy $E_{{\rm kin},j}^{\rm c}(t)$, owing to the density waves,
defined by

$E_{{\rm kin},j}^{\rm i}(t)=\frac{1}{2}\int\left[\left\\{\sqrt{n_{j}({\bf
r},t)}\nabla\theta_{j}({\bf r},t)\right\\}^{\rm i}\right]^{2}d{\bf r},$ (6)
$E_{{\rm kin},j}^{\rm c}(t)=\frac{1}{2}\int\left[\left\\{\sqrt{n_{j}({\bf
r},t)}\nabla\theta_{j}({\bf r},t)\right\\}^{\rm c}\right]^{2}d{\bf r}.$ (7)

Here, $\\{\cdots\\}^{\rm i}$ and $\\{\cdots\\}^{\rm c}$ denote the
incompressible and compressible components, respectively;
$\nabla\cdot\\{\cdots\\}^{\rm i}=0$ and $\nabla\times\\{\cdots\\}^{\rm c}=0$.
In the two-component system, the intercomponent-interaction energy $E_{{\rm
int},12}(t)$ is given by

$E_{{\rm int},12}(t)=g_{12}\int n_{1}({\bf r},t)n_{2}({\bf r},t)d{\bf r},$ (8)

and the total energy of the system is $E_{1}(t)+E_{2}(t)+E_{{\rm int},12}(t)$.

##  III Results

###  III.1 Vortex pair annihilation in two-component BECs

We consider the periods of the systems $L=16$, $32$, and $64$, and call these
systems the small, medium, and large system, respectively.

In the one-component system without dissipation, we confirm that the pair
moves with constant velocity and never gets annihilated as long as the cores
of two vortices do not overlap initially. Then, the incompressible and
compressible kinetic energy and quantum energy are almost constant with
fluctuation.

In the two-component system with $g_{12}\lesssim 0.8g$, we confirm that a
vortex pair also moves with constant velocity and never gets annihilated like
in the one-component system. In this paper, we show the results with
$g_{12}=0.9g$ to describe the annihilation of the vortex pair and the
recurrence.

Figure 1:  Dynamics when a vortex pair just gets annihilated in component 1 in
the large system of $L=64$ with $g_{12}=0.9g$. (a)–(f) describe component 1,
and (g)–(l) describe component 2. (a), (d), (g), and (j) are the density and
phase at $t=0$; (b), (e), (h), and (k) are those at $t=100$; and (c), (f),
(i), and (l) are those at $t=500$, respectively. The color bar of the density
is normalized by the uniform density, and the range is $0$ – $1.25$. The white
circles, ellipse, and arrows denote the vortex, antivortex, and rarefaction
pulse, respectively. Figure 2:  Time development of (a) $E_{j}(t)$ and
$E_{{\rm int},12}(t)$, (b) $E_{{\rm kin},1}^{\rm i}(t)$ and $E_{{\rm
kin},1}^{\rm c}(t)$, and (c) $E_{{\rm kin},2}^{\rm c}(t)$ in the large system
with $g_{12}=0.9g$. In (a), the red solid, blue dotted, and black dashed lines
show $E_{1}(t)$, $E_{2}(t)$, and $E_{{\rm int},12}(t)$, respectively. In (b)
and (c), the red solid and blue dotted lines denote $E_{{\rm kin},1}^{\rm
i}(t)$ and $E_{{\rm kin},j}^{\rm c}(t)$, respectively.

Figure 1 illustrates the density and phase at $t=0$, $t=100$, and $t=500$ of
the large system. The cores of the vortices of component 1 are filled by
component 2 (Fig. 1(a) and (g)) because the intercomponent interaction is
repulsive Eto  _et al._ (2011). In the initial state, only component 1 has a
vortex pair where the distance between the vortex and the antivortex is equal
to $5.75$. Then, density waves are not excited in any component, and thus,
$E_{{\rm kin},1}^{\rm i}(t)$ is dominant in the kinetic energy of the whole
system (Fig. 2(b) and (c)). Here, Fig. 2(c) only shows $E_{{\rm kin},2}^{\rm
c}(t)$ because $E_{{\rm kin},2}^{\rm i}(t)\simeq 0$. This pair moves upward
because of the self-induced velocity. As the pair moves, density waves are
emitted and the pair gets annihilated at $t\simeq 100$, and then a rarefaction
pulse appears (Fig. 1(b) and (e)). Thereafter, the rarefaction pulse decays
and the density waves are emitted. In these processes, $E_{{\rm kin},1}^{\rm
i}(t)$ is transferred not only to $E_{{\rm kin},1}^{\rm c}(t)$ but also to
$E_{{\rm kin},2}^{\rm c}(t)$ until $t\simeq 300$ (Fig. 2(b) and (c)) because
of the intercomponent interaction. Finally, the density waves spread in both
components (Fig. 1(c) and (i)). Then, the system becomes statistically steady
and vortex pairs never appear. Through the annihilation of a vortex pair and
the spread of the density waves, the phase of component 2 is almost uniform
(Fig. 1(j)–(l)).

As the vortex pair gets annihilated, the energies $E_{j}$ of both components
increase and the intercomponent-interaction energy $E_{{\rm int},12}$
decreases, while the sum of these total energies is conserved (Fig. 2(a)).
These increases and decreases are due to the spread of the density waves
emitted by the motion of the pair.

In the one-component system, the pair dose not get annihilated without
dissipation or a catalyst vortex. However, in the two-component system, the
pair can get annihilated even without dissipation or a catalyst vortex, which
is caused by the energy exchange through the intercomponent interaction. Here,
we describe the details of the exchange of energies. In the initial state, the
densities of both components are uniform except for the cores of the vortices,
which are represented by $n_{{\rm uni},j}$. The spread of the density waves
changes the densities as $n_{{\rm uni},j}+\delta n_{j}({\bf r},t)$. Here,

$\int\delta n_{j}({\bf r},t)d{\bf r}=0,$ (9)

because the total number of particles of both components are conserved. Then,
$E_{{\rm int},1}(t)$ can be calculated as

$\begin{array}[]{l}E_{{\rm int},1}(t)=\frac{g}{2}\int(n_{{\rm uni},1}+\delta
n_{1}({\bf r},t))^{2}d{\bf r}\\\ \\\ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{g}{2}\int
n_{{\rm uni},1}^{2}d{\bf r}+\frac{g}{2}\int\delta n_{1}({\bf r},t)^{2}d{\bf
r}\\\ \\\ \ \ \ \ \ \ \ \ \ \ \ \ \geq\frac{g}{2}\int n_{{\rm uni},1}^{2}d{\bf
r}.\end{array}$ (10)

Thus, the emission and spread of the density waves increase $E_{{\rm
int},1}(t)$.

In the two-component system, the energies $E_{j}$ of both components increase,
although the pair of component 1 gets annihilated. Although this seems
unusual, this is sustained in the two-component system. The annihilation of
the pair in component 1 excites the density waves in component 2 because of
the intercomponent interaction as well as those in component 1. Here, the
intercomponent interaction is repulsive; consequently, the local increase in
the density of component 1 pushes component 2 from there and vice versa to
yield $\delta n_{1}({\bf r},t)\delta n_{2}({\bf r},t)<0$ almost everywhere
(Fig. 1(c) and (i)). Then, $E_{{\rm int},12}(t)$ decreases compared with the
uniform system because

$\begin{array}[]{l}E_{{\rm int},12}(t)=g_{12}\int(n_{{\rm uni},1}+\delta
n_{1}({\bf r},t))(n_{{\rm uni},2}+\delta n_{2}({\bf r},t))d{\bf r}\\\ \\\ \ \
\ \ \ \ \ \ \ \ \ \ =g_{12}\int n_{{\rm uni},1}n_{{\rm uni},2}d{\bf
r}+g_{12}\int\delta n_{1}({\bf r},t)\delta n_{2}({\bf r},t)d{\bf r}\\\ \\\ \ \
\ \ \ \ \ \ \ \ \ \ \leq g_{12}\int n_{{\rm uni},1}n_{{\rm uni},2}d{\bf
r}.\end{array}$ (11)

Through this process, the total energy of the system is conserved, and $E_{1}$
and $E_{2}$ increase and $E_{{\rm int},12}$ decreases until $t\simeq 300$
(Fig. 2(a)), and then they become statistically steady.

###  III.2 Recurrence of a vortex pair

In the small system with $g_{12}=0.9g$, we find the creation of the
rarefaction pulse and the recreation of the vortex pair. Here, we call this
recreation the recurrence of the vortex pair. However, the recurrence does not
occur in the medium system with $g_{12}=0.9g$, although the rarefaction pulse
appears. These results are very different from those of the large system.

Figure 3:  Dynamics when a rarefaction pulse is created in both components in
the medium system of $L=32$ with $g_{12}=0.9g$. (a)–(d) describe component 1,
and (e)–(h) describe component 2. (a), (e), (c), and (g) are the density and
phase at $t=0$; (b) and (d) are those at $t=550$; and (f) and (h) are those at
$t=180$. The color bar of the density is normalized by the uniform density,
and the range is $0$ – $1.25$. The white ellipses and arrows denote the
rarefaction pulses. Figure 4:  Time development of (a) $E_{{\rm kin},1}^{\rm
i}(t)$ and $E_{{\rm kin},1}^{\rm c}(t)$, and (b) $E_{{\rm kin},2}^{\rm c}(t)$
in the medium system with $g_{12}=0.9g$. (a) shows the time development of the
kinetic energies of component 1 and (b) shows that of component 2. The
denotations of each line are same as those in Fig. 2

Figures 3 and 4 show the density, phase, and time development of the energies
in the medium system with $g_{12}=0.9g$. The pair gets annihilated at $t\simeq
100$ like in the large system, and $E_{{\rm kin},1}^{\rm i}(t)$ decreases
(Fig. 4(a)). Then, the density waves spread over the system. Thereafter, the
density waves focus on a narrow region and gradually develop into a
rarefaction pulse. During the process, $E_{{\rm kin},1}^{\rm c}(t)$ and
$E_{{\rm kin},1}^{\rm i}(t)$ increase until $t\simeq 550$, when the
rarefaction pulse appears clearly with $n_{1}({\bf r},t)\sim 10^{-2}$ and the
local phase changes $\pi$, as shown in Fig. 3(b) and (d). This rarefaction
pulse subsequently decays, and finally, the density waves just spread. In
component 2, the rarefaction pulse also appears at $t\simeq 180$, but $E_{{\rm
kin},2}^{\rm i}(t)$ shows a negligible increase (Fig. 4(b)) because the
rarefaction pulse is shallow ($n_{2}({\bf r},t)\sim 10^{-1}$) and the local
change in the phase is less than $\pi$ (Fig. 3(f) and (h)). Here, Fig. 4(b)
only shows $E_{{\rm kin},2}^{\rm c}(t)$ because $E_{{\rm kin},2}^{\rm
i}(t)\simeq 0$. This shallow rarefaction pulse decays. After these processes,
the system eventually enters the statistical steady state where the density
waves just spread in both components.

Figure 5:  Dynamics when a vortex pair is recreated in both components in the
small system of $L=16$ with $g_{12}=0.9g$. (a)–(d) describe component 1, and
(e)–(h) describe component 2. (a), (e), (c), and (g) are the density and phase
at $t=0$; (b) and (d) are those at $t=150$; and (f) and (h) are those at
$t=70$. The color bar of the density is normalized by the uniform density, and
the range is $0$ – $1.25$. The white circles and arrows denote the vortex and
antivortex, respectively. Figure 6:  Time development of (a) the number
$N_{\rm pair}(t)$ of vortex pairs, (b) $E_{{\rm kin},1}^{\rm i}(t)$ and
$E_{{\rm kin},1}^{\rm c}(t)$, and (c) $E_{{\rm kin},2}^{\rm i}(t)$ and
$E_{{\rm kin},2}^{\rm c}(t)$ in the small system with $g_{12}=0.9g$. The
denotations of each line are the same as those in Fig. 2

Figure 5 and 6 present the results of the small system with $g_{12}=0.9g$. The
rarefaction pulse is created in component 2 after the initial annihilation of
a pair in component 1, and this breaks up into two vortices immediately (Fig.
5(f) and (h), $t=70$). This pair also gets annihilated soon. After the
annihilation in component 2, another pair is recreated in component 1 (Fig.
5(b) and (d), $t=150$). In this way, two components keep creating a vortex
pair alternately. Figure 6(a) shows the time development of the number of
vortex pairs in each component, where a vortex is defined from the phase
profile as a topological defect with quantized circulation. When there is a
vortex pair in component $j$, $N_{{\rm pair},j}=1$, and this state appears
alternately between two components with a relatively long period (about
$150$). In each period, $N_{{\rm pair},j}(t)$ can change from unity to $0$ a
few times because a vortex pair can change continuously to a rarefaction pulse
with no topological singularity. The change from a pair to a rarefaction pulse
yields $N_{{\rm pair},j}=0$. The reverse change can occur, and thus, $N_{{\rm
pair},j}(t)$ alternates between unity and $0$ in a period. We call the
alternate creation of a vortex pair between two components the recurrence. The
recurrence is repeated several times (Fig. 6(a)).

The recurrence is also found in the characteristic time development of energy
components. Figure 7 shows the dynamics and time development of the kinetic
energies in component 2 from $t=0$ to $t=200$, and the first period of the
recurrence is included in the duration. Initially, $E_{{\rm kin},2}^{\rm
c}(t)$ increases and a rarefaction pulse appears at $t\lesssim 45$ (Fig. 7(a)
and (d)). Then, $E_{{\rm kin},2}^{\rm c}(t)$ starts to change to $E_{{\rm
kin},2}^{\rm i}(t)$ and a vortex pair is created around $t=70$ (Fig. 7(b) and
(d)). Finally, the vortex pair returns to a rarefaction pulse and $E_{{\rm
kin},2}^{\rm i}(t)$ changes to $E_{{\rm kin},2}^{\rm c}(t)$ around $t=100$
(Fig. 7(c) and (d)). Thereafter, both $E_{{\rm kin},2}^{\rm c}(t)$ and
$E_{{\rm kin},2}^{\rm i}(t)$ become small around $t=150$, and a vortex pair
appears in component 1 (Fig. 7(d) and 5(b) and (d)). These processes are
repeated alternately within each period in each component, and finally, the
recurrence slowly fades out and vanishes after $t\simeq 1000$.

Figure 7:  Dynamics and development of the kinetic energies within a period in
component 2 in the small system of $L=16$ with $g_{12}=0.9g$. (a)–(c) show the
densities at $t=45$, $t=70$, and $t=100$, respectively. (d) shows the time
development of $E_{{\rm kin},2}^{\rm i}(t)$ and $E_{{\rm kin},2}^{\rm c}(t)$
from $t=0$ to $t=200$. The vertical dashed black lines correspond to the time
frames shown in (a)–(c).

###  III.3 Distribution of the compressible and the incompressible kinetic
energy

In these systems, only component 1 has a vortex pair at the beginning.
However, the exchange of energies by the intercomponent interaction causes the
annihilation of the vortex pair and the spread of the density waves.
Thereafter, in the medium system, the rarefaction pulse is created and both
$E_{{\rm kin},1}^{\rm c}(t)$ and $E_{{\rm kin},1}^{\rm i}(t)$ increase
temporarily at about $t\simeq 550$, but decrease with the decay of the
rarefaction pulse (Fig. 4(a)). In the small system, the recurrence occurs;
then, $E_{{\rm kin},j}^{\rm c}(t)$ increases and changes to $E_{{\rm
kin},j}^{\rm i}(t)$ (Fig. 6(b) and (c)). Here, we show the spatial
distribution of $E_{{\rm kin},j}^{\rm c}(t)$ and $E_{{\rm kin},j}^{\rm i}(t)$
in Fig. 8 and 9 to confirm these processes.

Figure 8:  Distributions of the density (top row), compressible kinetic energy
$E_{{\rm kin},j}^{\rm c}$ (middle row), and incompressible kinetic energy
$E_{{\rm kin},j}^{\rm i}$ (bottom row) of component 1 of the medium system.
The left, center, and right columns show the results at $t=350$, $t=550$, and
$t=650$, respectively. The ranges of the color bars of each quantity are $0$ –
$1.25$ (density), $0$ – $0.25$ ($E_{{\rm kin},j}^{\rm c}$), and $0$ – $0.125$
($E_{{\rm kin},j}^{\rm i}$). Figure 9:  Distributions of the density (top
row), compressible kinetic energy $E_{{\rm kin},j}^{\rm c}$ (middle row), and
incompressible kinetic energy $E_{{\rm kin},j}^{\rm i}$ (bottom row) of
component 1 of the small system. The row data and the ranges of the color bars
are the same as those in Fig. 8. The left, center, and right columns show the
results at $t=100$, $t=120$, and $t=150$, respectively.

When the density waves spread out, the compressible kinetic energy is widely
distributed across the entire system. In the medium system, $E_{{\rm
kin},1}^{\rm c}$ is widely distributed (Fig. 8, $t=350$). Then, the values of
$E_{{\rm kin},1}^{\rm c}$ are relatively small everywhere. Over time, $E_{{\rm
kin},1}^{\rm c}$ returns to focus on a narrow region, creating the rarefaction
pulse ($t=550$). However, the change from $E_{{\rm kin},1}^{\rm c}$ to
$E_{{\rm kin},1}^{\rm i}$ is not very effective, and thus, the rarefaction
pulse does not develop into a vortex pair and just decays ($t=650$). In the
small system, the values of $E_{{\rm kin},j}^{\rm c}$ are relatively large
everywhere (Fig. 9, $t=100$). Thus, $E_{{\rm kin},j}^{\rm c}$ can concentrate
locally more than in the medium system ($t=120$), and the change from $E_{{\rm
kin},j}^{\rm c}$ to $E_{{\rm kin},j}^{\rm i}$ is sufficient to recreate a
vortex pair ($t=150$). This means that the smaller system causes a more active
exchange between the compressible and the incompressible kinetic energy. After
the recreation of a pair, $E_{{\rm kin},j}^{\rm c}$ concentrates at the vortex
cores and $E_{{\rm kin},j}^{\rm i}$ is distributed near the pair.

If the exchange of energy between $E_{{\rm kin},1}^{\rm i}$ and $E_{{\rm
kin},2}^{\rm i}$ is perfect, the recurrence should continue forever with some
constant period. In actual processes of recurrence, however, the exchange is
not perfect. Initially, $E_{{\rm kin},1}^{\rm i}$ is dominant in the kinetic
energy of the system and $E_{{\rm kin},j}^{\rm c}$ coming from stopping the
imaginary time evolution is negligible. Through the annihilation of a vortex
pair, $E_{{\rm kin},1}^{\rm i}$ changes to $E_{{\rm kin},1}^{\rm c}$, and the
compressible kinetic energy is exchanged between $E_{{\rm kin},1}^{\rm c}$ and
$E_{{\rm kin},2}^{\rm c}$ by the intercomponent interaction. Finally, $E_{{\rm
kin},2}^{\rm c}$ focuses on a narrow region and changes to $E_{{\rm
kin},2}^{\rm i}$ with creating a vortex pair in component 2, while some
$E_{{\rm kin},j}^{\rm c}$ remains in both components. This imperfection of
exchange between $E_{{\rm kin},1}^{\rm i}$ and $E_{{\rm kin},2}^{\rm i}$
occurs at every step of recurrence, and this leads to the lack of exact
periodicity in Fig. 6(a) and the eventual decay of recurrence.

When a vortex is created, the velocity of the super fluid exceeds the critical
velocity. The critical velocity is of the order of the sound velocity, which
comes from the dispersion relation, in one-component BECs Sasaki  _et al._
(2010). The dispersion relation is also calculated in two-component BECs
Pethick and Smith (2008), and the critical velocity of two-component BECs is
found to be of the order of the sound velocity. In this system, the superfluid
velocity becomes large in the narrow region $E_{{\rm kin},j}^{\rm c}$
concentrates. Thus, the maximum velocity of each component exceeds the sound
velocity in the medium or small system when a rarefaction pulse or a vortex
pair is created, but not in the large system. These results are consistent
with the estimation of the order of the critical velocity in two-component
BECs.

##  IV Conclusions

We find that the annihilation and the recurrence of vortex pairs occur in two-
component BECs without dissipation. Here, each phenomenon is sensitive to the
intercomponent-interaction parameter and the system size. If the
intercomponent interaction is weak, energy and annihilation exchange between
two components is negligible. However, a strong-enough intercomponent
interaction changes the situation drastically and annihilates a pair. The
medium system induces the concentration of the compressible kinetic energy
emitted by the annihilation, and then the rarefaction pulse is created.
However, the change from compressible to incompressible kinetic energy is not
very effective, and thus, a vortex pair is not created and the rarefaction
pulse just decays. The small system also induces the concentration of the
compressible kinetic energy emitted by the annihilation and changes it to the
incompressible kinetic energy. Then, the recurrence occurs several times.

In summary, for the recurrence to occur, the annihilation of the vortex pair
without dissipation and the concentration of the compressible kinetic energy
are required. The latter is certainly caused by the periodic boundary
condition and the small system size. However, the recurrence never occurs
without the strong repulsive intercomponent interaction, which causes the
annihilation of vortex pairs without dissipation and the exchange of energies
between two components. Hence, the recurrence is characteristic of two-
component BECs.

Here, the annihilation of vortex pair, which is required for the recurrence,
also depends on the initial distance between the vortices, and the distance
can be changed depending on when imaginary time evolution is stopped. In this
study, we confine the distance to $5.75$ and confirm that the annihilation
dose not occur when $g_{12}\lesssim 0.8g$. It is also confirmed that the
critical distance below which the annihilation occurs becomes small at the
limit of $g_{12}\rightarrow 0$, because the system becomes two independent
condensates at that limit one . Thus, the critical distance is the increasing
function of $g_{12}$. The more detailed dependence of the annihilation on the
distance is a future work.

We should comment on the time scale of a period of the recurrence in Fig. 7.
In component 2, $E_{{\rm kin},2}^{\rm i}$ increases and a rarefaction pulse
appears about $t\lesssim 45$, and a vortex pair returns to another rarefaction
pulse around $t=100$; the period of the recurrence is about $60$ in the early
time. The time scale of this period is comparable to the lifetime of the
vortex pair, because $E_{{\rm kin},1}^{\rm i}(t)$ immediately increases after
the vortex pair returns to a rarefaction pulse in component 2 around $t=100$
(Fig. 6(b) and (c)). The period just increases later around $t=500$ because of
the imperfection of the exchange between $E_{{\rm kin},1}^{\rm i}$ and
$E_{{\rm kin},2}^{\rm i}$, however, the order of the period is still similar.
The qualitative discussion of what parametrically sets this timescale would be
important.

###### Acknowledgements.

This work was supported by a Grant-in-Aid for JSPS Fellows (grant nos.
20J14459 and JP20H01855).

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