Hawking-like escape of the soliton from the trap in two-component Bose-Einstein condensate

Abstract

We demonstrate, that Bose-Einstein condensate can escape from the trap,
formed of combined linear periodic (optical lattice) and parabolic potentials,
and the escaping mechanism is similar to Hawking radiation from black hole. The
low-amplitude bright-bright soliton in two-component Bose-Einstein condensate
(where chemical potentials of the BEC first and second components are located
nearby the opposite edges of the first band of the optical lattice spectrum)
serves as an analogue of particle-antiparticle pair in Hawking radiation. It is
shown that parabolic potential, being applied to such two-component BEC, leads
to spatial separation of its components: BEC component with chemical potential
located in semi-infinite gap exerts the periodical oscillations, while the BEC
component, whose chemical potential is in the first finite gap, escapes from
the trap (due to negative effective mass of gap soliton). We also propose a
method for the creation of such bright-bright soliton -- transferring of atoms
from one BEC component to another by spatially periodic linear coupling term.

Full text

arXiv:1501.02401v1 [cond-mat.quant-gas] 10 Jan 2015
Hawking-like escape of the soliton from the trap in two-component Bose-Einstein
condensate
Yu. V. Bludov1∗and M. A. Garc´ıa- ˜
Nustes2
1Centro de F´ısica, Universidade do Minho, Campus de Gualtar, Braga 4710-057, Portugal
2Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Avenida Brasil, Valpara´ıso, Casilla 2950, Chile.
We demonstrate, that Bose-Einstein condensate can escape from the trap, formed of combined
linear periodic (optical lattice) and parabolic potentials, and the escaping mechanism is similar
to Hawking radiation from black hole. The low-amplitude bright-bright soliton in two-component
Bose-Einstein condensate (where chemical potentials of the BEC first and second components are
located nearby the opposite edges of the first band of the optical lattice spectrum) serves as an
analogue of particle-antiparticle pair in Hawking radiation. It is shown that parabolic potential,
being applied to such two-component BEC, leads to spatial separation of its components: BEC
component with chemical potential located in semi-infinite gap exerts the periodical oscillations,
while the BEC component, whose chemical potential is in the first finite gap, escapes from the
trap (due to negative effective mass of gap soliton). We also propose a method for the creation of
such bright-bright soliton – transferring of atoms from one BEC component to another by spatially
periodic linear coupling term.
PACS numbers: 03.75.Kk, 03.75.Lm, 67.85.Hj
I. INTRODUCTION
Hawking radiation is the remarkable property of black
hole: it occurs due to creation of a particle-antiparticle
pair nearby the black hole edge [1]. If one of the pair con-
stituents crosses the event horizon, it never returns, thus
giving rise to the emission from the black hole, which in
its turn leads to decreasing of black hole energy and mass
(for review see, e.g., Ref.[2]). Nevertheless, the main dif-
ficulty in the experimental confirmation of Hawking ra-
diation is the big mass of known black holes in universe,
and, hence, weak intensity of Hawking radiation. That
is why the reproducing of black holes in laboratory con-
ditions are of great interest. From one side, small arti-
ficial black holes can be created in large hadron collider
[3]. From the other side, the phenomena, originated by
gravity, are reproducible (through the similarity of wave
processes) in number of other physical systems, called
analogue gravity models [4]. In particular, black holes
and Hawking radiation can be obtained in electromag-
netic wave waveguide [5], slow light in moving medium
[6][32] or in optical fibers [8] (actually, first experimen-
tal evidence of Hawking radiation was observed in this
system [9], although the nature of observed phenomena
is also discussable [10]), surface waves on moving water
[11], the moving flow with a gradient from subsonic to
supersonic flow (so called acoustic or sonic black hole
[12]). Sonic black holes, after being predicted in 1981,
later was implemented in quantum liquids like liquid He
[13], atomic [14] and polaritonic [15] Bose-Einstein con-
densates (BEC). Moreover, reproducing in BEC a system
with inner horizon (analogue of charged black hole [16]),
it is possible to achieve experimentally a black hole las-
∗Electronic address: bludov@fisica.uminho.pt
ing, i.e. existence of self-amplifying Hawking radiation
[17].
One of factors, which determines a practical interest
to BEC (subjected to linear periodic potential – optical
lattice(OL)], is the possibility to use it as almost perfect
testbench, which allows to reproduce (without impurities
and defects) a lot of phenomena, known from condensed
matter (for review see, e.g. [18]). Nevertheless, due to
strong two-body interactions BEC is essentially nonlin-
ear system: the fact, which is hardly to ignore. A natural
way to take into account the nonlinearity is to consider
BEC matter waves in the form of localized wavepackets –
solitons, which can propagate at long distances without
loosing their shape. Moreover, loading the BEC in OL
allows one to control the dispersion (normal and anoma-
lous), and, as distinct from the case without OL, allows
the existence of solitons either in the case of attractive
as repulsive interaction (for review see Ref.[19]). Besides
the intensive theoretical study, the creation of bright gap
solitons in repulsively interacting BEC was demonstrated
experimentally [20].
Binary mixtures of Bose-Einstein condensates (BECs)
can support vector (multicomponent) solitons with va-
riety of interesting properties. The possibility for two-
component BEC to be loaded into OL was experimen-
tally demonstrated in Ref.[21]. Being loaded into the OL,
two-component BEC can sustain bright-bright and dark-
bright stationary solitons [22], mixed-symmetry modes
and breathers [23]. Nevertheless, experimental study of
vector solitons [24] was restricted only to the case of dark-
bright solitons in two-component BEC without OL.
In the present paper we consider emission of mat-
ter wave solitons from two-component BEC, loaded into
parabolic trap with optical lattice. We use periodic sign-
varying linear coupling term to create vector soliton,
which is characterized by first- and second-component
chemical potentials located in the vicinity of the lower

2
and upper edges of the OL spectrum first band, respec-
tively. After being loaded into the parabolic trap, the
first component of vector soliton (characterized by posi-
tive effective mass nearby the bottom edge of first band)
oscillates periodically in the trap, while second compo-
nent (characterized by negative effective mass at top edge
of the first band), is accelerated by parabolic trap and
escapes from it. In general, the nature of investigated
phenomena is similar to Hawking emission, but at the
same time has significant differences. Firstly, in our
case the vector soliton with opposite masses of compo-
nents (analogue of particle-antiparticle pair) is created
by transferring atoms from first to second components
of BEC. In conventional Hawking emission, by contrast,
the particle-antiparticle pair is formed from vacuum due
to quantum fluctuations. Secondly, in our phenomenon
the event horizon is defined somehow artificially (as an
edge of potential well). That is why we prefer to call
our phenomenon Hawking-like emission (in analogy with
Ref.[25]).
The paper is organized as follows. In Sec.II we ex-
pose and describe the model of our problem (problem
statement). In Sec.III we represent the stationary state
of bright-bright solitons in the two-component BEC and
propose a mechanism to create this state. In Sec.IV
we describe the dynamics of two-component soliton in
parabolic trap and the mechanism of Hawking-like es-
cape of the soliton from the trap.
II. THE MODEL AND PRELIMINARY
ARGUMENTS
To be specific, we consider a spinor BEC composed of
two hyperfine states, say of the |F= 1, mf=−1iand
|F= 2, mf= 1istates of 87 Rb atoms [26] confined at dif-
ferent vertical positions by transverse parabolic traps and
loaded into the optical lattice of cos-like shape. Addition-
ally, a time-dependent external potential γ(x, t) (which
is aperiodic in general case) is applied to the condensate.
At the same time hyperfine states are coupled by a coor-
dinate and time-dependent coupling field β(x, t), which
describes the possibility of conversion of atoms between
states (such kind of coupling can be originated by the
external magnetic field).
We assume the condensate to be quasi-one-dimensional
(cigar-shaped). Then, in the mean-field approximation
the system is described by the GP equations [27, 28]
i∂ψ1
∂t =−∂2ψ1
∂x2−Vcos(2x)ψ1+γ(x, t)ψ1+
β(x, t)ψ2+g1|ψ1|2+g|ψ2|2ψ1,(1a)
i∂ψ2
∂t =−∂2ψ2
∂x2−Vcos(2x)ψ2+γ(x, t)ψ2+
β(x, t)ψ1+g|ψ1|2+g2|ψ2|2ψ2.(1b)
Equations (1) are written in dimensionless form: all en-
ergy values (like OL amplitude V) are measured in re-
coil energy units of ER=~2π2/(2md2) (where mis the
atomic mass and dis the OL period), while the coordi-
nate xand time tare measured in units of d/π and ~/ER,
respectively. At the same time wavefunctions ψj(x) are
measured in the a2
⊥π2/(4d2|a12|) units, where a⊥is the
transverse [in the (y, z )-plane] oscillator length, a12 is
the inter-species s-wave scattering length [thus, g=±1
in (1)].
In the absence of nonlinearity (formally g , g1, g2≡
0), additional external force γ(x, t)≡0 and cou-
pling term β(x, t)≡0, periodicity of OL gives rise to
band-gap structure of both spectrum En(q) and Bloch
functions ϕnq(x) of the linear problem En(q)ϕnq (x) =
−d2ϕnq/dx2−Vcos(2x)ϕnq(x), where nis the band num-
ber and qis the Bloch wavenumber in the first Brillouin
zone, q∈[−1,1]. Meanwhile both the spectrum and
the Bloch functions are periodic in the reciprocal space
with period 2 (in the chosen units): En(q) = En(q+ 2),
ϕn,q(x) = ϕn,q+2(x). In further considerations we take
into account the first band only (n= 1), whose bottom
and top correspond to E1(0) and E1(1), respectively, so
the index nwill be omitted.
III. SMALL-AMPLITUDE COUPLED
SOLITONS.
A. Stationary state
Considering solutions of Eqs.(1) [in the absence of
additional external potential γ(x, t)≡0 and coupling
β(x, t)≡0] and limiting ourself to the small amplitude
case, we introduce small parameter ε. Following this ap-
proach, we consider that the chemical potential of the
first component soliton µ1to be in the semiinfinite gap
[µ1< E(0)], while chemical potential of second com-
ponent soliton µ2is in the first finite gap [µ2> E(1)].
Then the wavefunctions can be approximated by ψj≈
εAj(ξ, τ )ϕj−1(x) exp (−iE(j−1)t) where Aj(ξ , τ) is an
amplitude depending on slow variables ξ=εx,τ=ε2t,
and ϕq(x) is the Bloch wavefunction on top (q= 1) or
bottom (q= 0) of the band. We observe, that by choos-
ing εto be the small parameter we implicitly impose
the conditions where the characteristic scale of the ex-
citations is determined by the detuning of the chemical
potential towards the adjacent gap.
Using the standard algebra (the details can be found
say in Ref. [29]) one verifies that Aj(ξ , τ) solve the cou-
pled nonlinear Schr¨odinger equations
i∂A1
∂τ =−1
2M0
∂2A1
∂ξ2+g1χ0|A1|2+gχ|A2|2A1,(2a)
i∂A2
∂τ =−1
2M1
∂2A2
∂ξ2+gχ|A1|2+g2χ1|A2|2A2.(2b)
Here Mq= (d2E(q)/dq2)−1is the effective mass, χq=
Rπ/2
−π/2|ϕq(x)|4dx,χ=Rπ/2
−π/2|ϕ0(x)ϕ1(x)|2dx.

3
FIG. 1: (Color online) (a) Relation (6) [red solid line] be-
tween second-component chemical potential µ2and the first-
component one µ1of the stationary coupled solution (5) and
chemical potential µ1,i of the initial stationary state versus
µ1of the stationary coupled solution (5) [blue dashed line];
(b) First and second component bifurcation diagrams N1(µ1)
and N2(µ2), corresponding to the stationary coupled solution
(5); (c,d) Shapes of coupled solitons (5) with chemical poten-
tials µ1=−0.938, µ2=−0.7317 (soliton norms N1= 0.0085,
N2= 0.0366, respectively) – first ψ1(x, 0) and second ψ2(x, 0)
components are depicted in panels (c) and (d), respectively.
In all panels the parameters of OL and nonlinearities are:
V= 3.0, g1=−2, g2= 2, g=−1. The bottom and right
dashed regions in panel (a) as well as dashed region in the
middle of panel (b) are referred to the first band of the OL
spectrum.
When g1<0, g2>0, Eqs.(2a)–(2b) pos-
sess the stationary solution in the form Aj=
ajcosh−1(p−2Mj−1Ωjξ) exp(−iΩjτ), where
a2
1= 2 Ω2gχ −Ω1|g2|χ1
|g1g2|χ0χ1+χ2,(3)
a2
2= 2 Ω1gχ + Ω2|g1|χ0
|g1g2|χ0χ1+χ2,(4)
and M0Ω1=M1Ω2<0 (solitons in both component pos-
sess equal width), thus implying Ω1<0 and Ω2>0. As
a result, the particular solution of (1), representing cou-
pled soliton stationary state ψj(x, t) = φj(x) exp(−iµjt)
(j= 1,2) will be written as
φ1(x, t) = 2[µ1−E(0)]
M1
M0gχ −M1|g2|χ1
|g1g2|χ0χ1+χ21/2
×
ϕ0(x)
cosh(p−2M0(µ1−E(0))x),(5a)
φ2(x, t) = 2[µ2−E(1)]
M0
M1gχ +M0|g1|χ0
|g1g2|χ0χ1+χ21/2
×
ϕ1(x)
cosh(p−2M1[µ2−E(1)]x).(5b)
Additionally, the relations between µ1,2and Ω1,2, namely
µ1=E(0) + ε2Ω1,µ2=E(1) + ε2Ω2, imply the linear
character of dependence between chemical potentials of
the first and second components
µ2(µ1) = E(1) + M0
M1
[µ1−E(0)] ,(6)
an example of which for certain parameters of OL and
nonlinearities is depicted in Fig.1(a) [red solid line].
Thus, in the case under consideration chemical poten-
tials µ1,2are not independent variables: detuning of the
first-component chemical potential from the bottom band
edge E(0) > µ1results into the simultaneous detuning of
second-component chemical potential from the top edge
of the band E(1) < µ2(towards the center of the first
gap).
The soliton norm (N1,2=R|φ1,2|2dx) in each compo-
nent of the stationary state, namely
N1(µ1) = 23/2
π|M1|E(0) −µ1
M01/2M0gχ −M1|g2|χ1
|g1g2|χ0χ1+χ2,(7)
N2(µ2) = 23/2
πM0µ2−E(1)
|M1|1/2M1gχ +M0|g1|χ0
|g1g2|χ0χ1+χ2,(8)
can be calculated directly from (5), substituting rapidly-
oscillating Bloch functions ϕ0,1(x) by their average value
on the period ϕ2
0,1(x)=π−1. The results are depicted
in Fig.1(b) in the form of the bifurcation diagrams, show-
ing the growth of the soliton norm of both components
N1,2with an increase of the detuning of chemical poten-
tials from respective band edges. Examples of the soliton
shapes are represented in Fig.1(c),1(d), showing different
symmetry of coupled soliton stationary state components
(in its turn determined by the symmetry of the respec-
tive Bloch functions). In particular, while the first com-
ponent (referring to the semi-infinite gap soliton) is the
sign-constant, the second component (referring to the fi-
nite gap soliton) is the sign-alternating.
B. Creation of the stationary coupled state.
Now the natural question arises: how to create the
coupled soliton stationary state? In order to answer this
question, we introduce the concept of the initial station-
ary state: the particular solution of (1) with all atoms
concentrated in the first component, namely
ψ1,i(x, t) = 2[E(0) −µ1,i]
|g1|χ01/2
×
ϕ0(x) exp{−iµ1,it}
cosh(p−2M0(µ1,i −E(0))x),(9a)
ψ2,i(x, t) = 0.(9b)
As the matter of fact, in the initial stationary state atoms
(in the first component) are localized into bright soliton,

4
FIG. 2: (Color online) (a) Exposure time, necessary to achieve
state with soliton norms in first and second components N1=
0.0085, N2= 0.0366 respectively versus coupling amplitude
Vc; (b–d) Temporal evolution of projections |c1,2(t)|2(panel
b) and spatio-temporal evolution of wavefunctions |ψ1(x, t)|2
(panel c) and |ψ2(x, t)|2(panel d) of two-component BEC
with linear coupling of the form (13) with Vc= 0.25, T1= 10,
T2= 16.12 (depicted by two dashed lines in panels b–d).
In all panels other parameters are the same as in Fig.1, and
initial stationary state is characterized by the first component
chemical potential µ1,i =−0.94 and soliton norm N1,i =N1+
N2.
whose shape is sign-constant [similar to one, depicted in
Fig.1(c)] and soliton norm can be expressed as
N1,i(µ1,i ) = 23/2
π|g1|χ0E(0) −µ1,i
M01/2
,(10a)
N2,i = 0.(10b)
Since the system of coupled nonlinear Schr¨odinger equa-
tions (1) is conservative, to create a coupled soliton sta-
tionary solution with solitons norms N1and N2, one
should start from the initial stationary state with soli-
ton norm in first component equal to
N1,i(µ1,i ) = N1(µ1) + N2(µ2).(11)
The respective relation between the initial stationary
state chemical potential µ1,i and the value of µ1[which
satisfies the condition (11)] is depicted in Fig.1(a) by
blue dashed line. As the next step, applying nonzero
linear coupling between components β(x, t)6= 0 for cer-
tain time, one can transfer the atoms from the first com-
ponent to the second one. Nevertheless, application of
spatially uniform (coordinate independent) coupling will
result in the transferring of the atoms to second compo-
nent in the same phase, giving rise to the similar shapes of
solitons in both components (in fact, such resulting soli-
ton will be unstable due to opposite signs of intraspecies
nonlinearities g1and g2). Overcoming of this difficulty
and creation of the coupled soliton stationary state with
different symmetries in the first and second components
[Figs.1(c),1(d)] is possible by using the spatially-periodic
linear coupling with the period equal to 2π, namely
β(x, t) = Vccos(x).(12)
Here Vcis the amplitude of coupling.
If start with the initial stationary state, the exposure
time T, which is necessary to achieve the coupled soliton
stationary state [depicted in Figs.1(c),1(d)] is represented
in Fig.2(a) as function of the coupling amplitude Vc. As it
is evident, there exist a certain amplitude threshold [des-
ignated in Fig.2(a) by vertical dashed line]: the desired
coupled soliton stationary state can be achieved only for
the values of Vcabove this threshold. Above the thresh-
old, increasing of coupling amplitude results into the de-
creasing of the exposure time.
As a result, the creation of the coupled soliton station-
ary state can be achieved by switching on the coupling at
time moment T1and switching it off after exposure time
T=T2−T1, namely
β(x, t) = 


0, t < T1
Vccos(x), T1< t < T2
0, t > T2
(13)
The creation of coupled soliton stationary state is evident
from Fig.2(b), which represents the projections of the
wavefunctions ψj(x, t) (j= 1,2) on the correspondent
stationary state φj(x) [see Eq.(5)], i.e.
cj(t) = 1
NjZ∞
−∞
ψj(x, t)φj(x)dx. (14)
One observes that after coupling is switched off at time
moment T2, square modula of these projections are ap-
proximately equal to unity during the relatively long in-
tegration time. In more details process of the creation of
coupled soliton stationary state is depicted in Figs.2(c)
and 2(d), which demonstrate both the stability of the
initial stationary state at t < T1, and the stability of the
created coupled soliton stationary state at t > T2.
IV. HAWKING-LIKE EMISSION OF MATTER
FROM THE POTENTIAL WELL
When small external force γ(x, t) is switched on, the
dynamics of solitons in real and reciprocal spaces is de-
scribed by the semiclassical equations [30]
˙
Xj=dE
dq 


q=Qj
,˙
Qj=−∂γ(x, t)
∂x 


x=Xj
,(15)
where Xjand Qjdenote the center of mass of the j-
component soliton in real and reciprocal space, respec-
tively, and the overdot stands for the time derivative.
When the parabolic time-independent potential
γ(x, t) = ν(x−x0)2(16)

5
FIG. 3: (Color online) Spatio-temporal evolution of particle
density |ψ1(x, t)|2[panel (a)] and |ψ2(x, t)|2[panel (b)] (de-
picted by color maps) in two-component BEC with nonlin-
earities g1=−2, g2= 2, g=−1 and loaded into OL with
amplitude V= 3.0 and parabolic trap (16) with ν= 5 ·10−5,
x0= 10. The initial condition is the stationary gap soliton
with µ1=−0.938, µ2=−0.7317, [depicted in Figs.1(c),1(d)].
Soliton center position in real space X1(t) and X2(t) [obtained
from Eqs.(19) and (21) with the above-mentioned parameters]
are depicted by white lines in panels (a) and (b), respectively,
while soliton center positions in the reciprocal space Q1(t)
and Q2(t) [see Eqs.(20),(22)] are represented in lateral fig-
ures. The correspondent shape of the parabolic trap γ(x) is
presented in two upper figures.
is applied to the two-component soliton, Eq.(15) can be
written as
Xj=x0−˙
Qj
2ν,(17a)
¨
Qj+ 2νω1πsin (πQ) = 0.(17b)
In Eqs.(17) we approximated band structure E(q) leaving
only two leading terms of Fourier expansion, i.e.
E(q) = ω0+ω1cos(πq).(18)
Eq.(17b) is the differential equation, which describes the
oscillation of simple pendulum. Thus, under initial condi-
tions X1(0) = X2(0) = 0, Q1(0) = 0, Q2(0) = 1, system
of Eqs.(17) possesses an exact solution
X1(t) = x01−cn πνx0
k1
t, k1,(19)
Q1(t) = 2
πarcsin k1sn πνx0
k1
t, k1,(20)
X2(t) = x0
1−1
dn π√2νω1
k2t, k2
,(21)
Q2(t) = 2
πam π√2νω1
k2
t, k2.(22)
In the above equations
k1=νx2
0
2ω11/2
, k2=1 + νx2
0
2ω1−1/2
,
are elliptic moduli, cn ( t, k), dn (t, k) are the Jacobi ellip-
tic functions, am (t, k) is the Jacobi amplitude.
As it follows from Eqs.(20) and (22), in the recipro-
cal space soliton center of first component should exhibit
periodical oscillations in the vicinity of Q1= 0 [see the
lateral panel of Fig.3(a)], and the soliton center in the
second component [the lateral panel of Fig.3(b)] is the
increasing function of time [33]. At the same time in coor-
dinate space soliton centers of both first and second com-
ponents exhibit periodic oscillations [see Eqs.(19) and
(21)]. Nevertheless, while first component of soliton os-
cillates in the vicinity of the parabolic trap minimum
x0, with the oscillation period τ1= 4K(k1)(2π2νω1)−1/2
and amplitude ξ1=x0[white lines in Fig.3(a)], the soli-
tonic second component moves in the region x < 0 (and
even does not reach the trap center x0) with the pe-
riod τ2= 2k2K(k2)(2π2νω1)−1/2and amplitude ξ2=
(p(2ω1+νx2
0)/ν −x0)/2 [white lines in Fig.3(b)].
These predictions are confirmed by direct numerical
integration of Eqs.(1), the spatio-temporal evolution of
two-component soliton being depicted by color maps in
Fig.3. Nevertheless, while periodical oscillations of soli-
tonic first component are stable [soliton keeps its shape
during long time of evolution, see Fig.3(a)], in the sec-
ond component soliton is destroyed [Fig.3(b)] after cer-
tain time, less than one oscillation period τ2. This phe-
nomenon can be explained in the following manner: in
the first component soliton oscillates in the narrow in-
terval in reciprocal space ∼ −0.03 ≤Q1≤∼ 0.03 [see
lateral panel in Fig.3(a)], inside which effective mass is
always positive, MQ1>0. As a result, the modulational
instability condition MQ1g1<0 is kept at every mo-
ment of time, thus preventing the soliton from the de-
terioration. Contrary, in the second component Q2[lat-
eral panel in Fig.3(b)] passes through all values inside
the first band of the spectrum, even where the instabil-
ity condition MQ2g2<0 is not met, which result in the
destruction of soliton. Nevertheless, comparison of nu-
merical results with semiclassical ones shows good cor-
respondence between predicted amplitudes of oscillation
in coordinate space [Figs.3(a) and 3(b)] and reasonable
correspondence between oscillation periods. The discrep-
ancy between periods of oscillations takes place both due
to the inexactness of the band approximation (18) and
due to inexactness of semiclassical equations (15) in the
nonlinear case.
At the initial stage of evolution t&0 the positions of
the soliton in reciprocal space Q1and Q2increase due
to negative dγ(x)/dx at x= 0 [as it follows from (15)].
In its turn positive dE(q)/dq at q= 0 + 0 causes first-
component soliton motion to the positive direction of x-
axis, while negative derivative dE(q)/dq at q= 1 + 0
is responsible for the motion of second component to

6
the negative direction of x-axis (where dγ(x)/dx < 0),
i.e. outwards the parabolic trap center (similar result for
the one-component soliton was demonstrated in Ref.[31]).
So, it is natural to presuppose, that if second compo-
nent soliton is accelerated during the initial stage, and
at certain coordinate xEH the action of external trap
is switched off, i.e., dγ(x)/dx = 0, then it will con-
tinue its motion with constant velocity, escaping from
the parabolic trap. This coordinate xE H can be con-
sidered as analogue of event horizon in black hole. In
general, the coordinate xEH should be less than the po-
sition of second-component soliton center in real space at
quarter-period X2(τ2/4). This requirement comes from
the necessity to stop action of external force, when the
second-component soliton center in the reciprocal space
is inside the interval 1 < Q2<1.5 (where effective mass
is negative) in order to prevent soliton from further de-
struction.
As an example, this idea can be realised, when the soli-
ton is placed inside the finite-width parabolic potential
[compare with Eq.(16)]
γ(x, t) = ν(x−x0)2,|x−x0|< L/2,
ν(L/2−x0)2,|x−x0| ≥ L/2,(23)
where Lis the width of the potential well. The shape
of the potential (23) is depicted in the upper panels of
Fig.4. In the frame of the above-mentioned formalism,
edges of the finite-width potential (23), xE H1=x0−L/2
and xEH2=x0+L/2, can be considered as event hori-
zon analogues. As seen from Figs.4(a) and 4(b), first-
component soliton oscillates periodically [see Fig.4(a)]
inside the finite-width potential well (23). Meanwhile,
second-component soliton is accelerated during its move-
ment in the negative direction of x-axis [see Fig.4(b)], and
after crossing the potential-well edge xEH1(depicted by
vertical dash-and-dotted line) continues its motion with
constant velocity.
Moreover, we can start from the situation when all the
BEC atoms are initially concentrated in the first com-
ponent [like Eq.(9)], and then apply (for the finite time)
the spatially-periodic linear coupling, which will transfer
a portion of atoms to the second component. In other
words, we use the same method, as described in Sec.III,
but apply it not to stationary soliton, but to the soli-
ton oscillating inside the finite-width potential well [see
Fig.4(c)]. In this case atoms, transferred to the second
component, will constitute the gap soliton with nega-
tive effective mass, which in its turn will escape from the
finite-width potential well, as demonstrated in Fig.4(d).
V. CONCLUSIONS
Thus, we described a mechanism of stimulated emis-
sion of matter waves (in form of bright solitons) from
the two-component BEC, loaded into the OL, which is
combined with the external parabolic potential. The
similarity between the Hawking emission from the black
FIG. 4: (Color online) Spatio-temporal evolution of particle
density |ψ1(x, t)|2[panels (a) and (c)] and |ψ2(x, t)|2[pan-
els (b) and (d)], obtained from the numerical integration of
Eqs.(1) with finite-width parabolic trap (23) and with the
linear coupling term β(x, t) = 0 [panels (a) and (b)], or of
the form (13) [panels (c) and (d)] with Vc= 0.25, T1= 500,
T2= 506.12 (exposure time is the same as in Fig.2). The ini-
tial conditions [in panels (a) and (b)] corresponds to that in
Fig.3, and [in panels (c) and (d)] are the same as in Fig.2. In
all panels the parameters of OL, finite-width parabolic trap,
and nonlinearities are V= 3.0, ν= 5·10−5,x0= 10, L= 80π,
g1=−2, g2= 2, g=−1. In all panels edges of the finite-
width parabolic trap are depicted by vertical dash-and-dotted
lines (the correspondent shape of the finite-width parabolic
trap γ(x) is presented in two upper figures). In panels (c)
and (d) time moments T1and T2are depicted by horizontal
dashed lines (indistinguishable in the scale of panels).
hole and the soliton escape from the parabolic trap is de-
fined by the fact, that we use the bright-bright soliton,
where chemical potentials of first and second BEC com-
ponent lie nearby the opposite edges of the first band
of OL spectrum. As a consequence, signs of the ef-
fective masses (which characterize the BEC first and
second components) are also opposite, and such type
of low-amplitude soliton can be considered as an ana-
logue of the particle-antiparticle pair in Hawking emis-
sion. We demonstrated, that this low-amplitude bright-
bright two-component soliton can be created by partial
transferring of atoms from one to another BEC compo-
nent, using spatially-periodic linear coupling term, whose
period equals to the double OL period. Being loaded into
the finite-width parabolic trap, one component of such
bright-bright soliton (characterized by the positive effec-

7
tive mass) exerts periodic oscillations nearby the trap
center, while another component (with negative effective
mass) is gradually accelerated and moves in the direc-
tion of parabolic trap growth. If soliton with negative
effective mass passes the finite-width parabolic trap edge
(which is the event horizon analogue), it escapes from the
trap and never returns to the initial point.
Acknowledgment
M.A.G.-N. thanks for the financial support of FONDE-
CYT project 11130450.
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