Ninth International Conference on
Geometry, Integrability and Quantization
June 8–13, 2007, Varna, Bulgaria
Ivaïlo M. Mladenov, Editor
SOFTEX, Sofia 2008, pp 224–240
Geometry,
Integrability
and
IX
Quantization
BOSE-FERMI MIXTURES IN TWO OPTICAL LATTICES
NIKOLAY A. KOSTOV, VLADIMIR S. GERDJIKOV
AND TIHOMIR I. VALCHEV
Institute for Nuclear Research and Nuclear Energy
Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
Abstract. We present stationary and travelling wave solutions for equations
describing Bose-Fermi mixtures in an external potentials which are elliptic
functions of modulus k. There are indications that such waves and localized
objects may be observed in experiments with cold quantum degenerate gases.
1. Introduction
Recently, there has been a strong interest on quantum degenerate mixtures of
bosons and fermions [3, 14, 16]. In this paper, we study a system of coupled nonlinear Schrödinger equations modelling a quantum degenerate mixture of bosons
and fermions in optical lattice. Here we extend the results of our recent paper [10]
and obtain new exact solutions in elliptic functions for the case when the boson and
fermion ingredients are trapped by potentials with different strengths V0,F 6= V0,B .
2. Bose-Einstein Mixtures in Optical Lattice: Basic Equations in
Mean Field Approximations
In this section we consider a mixture of BEC consisting of one boson and Nf
fermion ingredients. In the one-dimensional approximation it is described by the
following Nf + 1 coupled equations (see [16] and the references therein)
i~
∂Ψb
1 ∂ 2 Ψb
+
− VB Ψb − gBB |Ψb |2 Ψb − gBF ρf Ψb = 0
∂t
2mB ∂x2
f
∂Ψfj
1 ∂ 2 Ψj
i~
+
− VF Ψfj − gBF |Ψb |2 Ψfj = 0
∂t
2mF ∂x2
224
(1)
(2)
225
Bose-Fermi Mixtures in Two Optical Lattices
where ρf =
PNf
f 2
i=1 |Ψi |
gBB
and
2aBB
=
,
as
gBF
2aBF
=
,
as α
s
as =
~
·
mB ω⊥
(3)
aBB and aBF are the scattering lengths for s-wave collisions for boson-boson and
boson-fermion interactions, respectively. An appropriate class of periodic potentials to model the quasi-1D confinement produced by a standing light wave is given
by [4]
VB = V0,B sn2 (αx, k),
VF = V0,F sn2 (αx, k)
(4)
where sn(αx, k) denotes the Jacobian elliptic sine function [2] with elliptic modulus 0 ≤ k ≤ 1.
Experimental realization of two-component Bose-Einstein condensates have stimulated considerable attention in the quasi-1D regime [7] when the Gross-Pitaevskii
equations for two interacting Bose-Einstein condensates reduce to coupled nonlinear Schrödinger (CNLS) equations with an external potential. In specific cases
the two component CNLS equations [1, 9, 13] can be reduced to the Manakov system [12] with an external potential. Elliptic solutions for the CNLS and Manakov
system were derived in [6, 8, 15].
In the presence of external elliptic potential explicit stationary solutions for NLS
were derived in [4, 5]. These results were generalized to the n-component CNLS
in [7]. For two component CNLS explicit stationary solutions are derived in [11].
3. Type A Travelling Wave Solutions with Non-Trivial Phases
At first we restrict our attention to stationary solutions of these CNLS
Ψb (x, t) = q0 (x) e−i
ω0
t+iΘ0 (x)+iκ0
~
ω
−i ~j
Ψfj (x, t) = qj (x) e
(5)
t+iΘj (x)+iκ0,j
(6)
where j = 1, . . . , Nf , κ0 , κ0,j , are constant phases, qj and Θ0 , Θj (x) are realvalued functions connected by the relation
Θ0 (x) = C0
Z x
dx0
0
q02 (x0 )
,
Θj (x) = Cj
Z x
dx0
0
qj2 (x0 )
·
(7)
C0 , Cj , j = 1, . . . , Nf being constants of integration. Substituting the Ansatz (5),
(6) in Equation (1) and separating the real and imaginary part we get

Nf

X
1
1 C02
q0xx − gBB q03 − VB q0 − gBF  qi2  q0 + ω0 q0 =
2mB
2mB q03
i=1
(8)
226
Nikolay A. Kostov, Vladimir S. Gerdjikov and Tihomir I. Valchev
1
1 Cj2
qjxx − gBF q02 qj − VF qj + ωj qj =
·
2mF
2mF qj3
(9)
We seek solutions for q02 and qj2 , j = 1, . . . , Nf as a quadratic function of sn(αx, k)
q02 = A0 sn2 (αx, k) + B0 ,
qj2 = Aj sn2 (αx, k) + Bj .
(10)
Equating the coefficients of equal powers of sn(αx, k) results in the following
relations among the solution parameters ωj , Cj , Aj and Bj and the characteristic
of the optical lattice V0 , α and k
Nf
X
Aj =
α2 k 2
gBF
ω0 =
X
α 2 k 2 B0
α2 (k 2 + 1)
Bi +
+ gBB B0 + gBF
2mB
2mB A0
i=1
j=1
1
gBB
−
mB mF gBF
1
−
gBF
V0B −
V0F gBB
gBF
(11)
Nf
α2 (k 2 + 1)
α 2 k 2 Bj
ωj =
+ gBF B0 +
,
2mF
2mF Aj
α2 k 2 − mF V0F
A0 =
mF gBF
(12)
α 2 B0
α 2 Bj
(A0 +B0 )(A0 +B0 k 2 ),
Cj2 =
(Aj +Bj )(Aj +Bj k 2 ) (13)
A0
Aj
where j = 1, . . . , Nf . Next for convenience we introduce
C02 =
B0 = −β0 A0 ,
Bj = −βj Aj ,
j = 1, . . . , Nf
(14)
then
C02 = α2 A20 β0 (β0 − 1)(1 − β0 k 2 )
(15)
Cj2
(16)
=α
2
A2j βj (βj
2
− 1)(1 − βj k ).
In order that our results (10) are consistent with the parametrization (5), (6), (7) we
must ensure that both q0 (x) and Θ0 (x) are real-valued, and also qj (x) and Θj (x)
are real-valued; this means that C02 ≥ 0 and q02 (x) ≥ 0 and also Cj2 ≥ 0 and
qj2 (x) ≥ 0. An elementary analysis shows that one of the following conditions
1
(17)
k2
for l = 0, . . . , Nf must hold. Using the well known transformation x → x − cj t,
j = 0, . . . , Nf it is easy to obtain travelling wave solutions with different velocities
cj
a)
Al ≥ 0,
βl ≤ 0
Ψb (x, t) = q0 (x − c0 t) e−i
Ψfj (x, t) = qj (x − cj t) e−i
b)
ω0
t−i~
~
ωj
~
Al ≤ 0,
1 ≤ βl ≤
mB ( 12 c20 t+c0 x)+iΘ0 (x)+iκ0
t−i~ mB ( 12 c2j t+cj x)+iΘj (x)+iκ0,j
227
Bose-Fermi Mixtures in Two Optical Lattices
where j = 1, . . . , Nf , κ0 , κ0,j , are constant phases, qj and Θ0 , Θj (x) are realvalued functions connected by the relation
Θ0 (x, t) = C0
Z x−c0 t
dx0
q02 (x0 )
0
,
Θj (x, t) = Cj
Z x−cj t
dx0
0
qj2 (x0 )
,
j = 1 . . . , Nf .
We display also mixed type solutions for which the boson part has trivial phase
while the fermions have nontrivial phases and vice versa. These are obtained with
1. generic Cj and B0 = −A0 , B0 = 0 or B0 = −A0 /k 2 .
2. C0 generic and Bj = −Aj , B0 = 0 or Bj = −Aj /k 2 .
Under certain conditions Qj (x, t) become periodic functions of x, see [10, 11]. If
the periods T0 , Tj satisfy
Θ0 (x + T0 ) − Θ0 (x) = 2πp0 ,
Θj (x + Tj ) − Θj (x) = 2πpj
(18)
for j = 1, . . . , Nf then Ψb , Ψfj will be periodic in x with periods T0 = 2m0 ω/α,
Tj = 2mj ω/α. This holds true provided there exist pairs of integers m0 , p0 , and
mj , pj , such that
m0
mj
= −π [αv0 ζ(ω) + ωτ0 /α]−1 ,
= −π [αvj ζ(ω) + ωτj /α]−1
p0
pj
where ω (and ω 0 ) are the half-periods of the Weierstrass functions ζ.
When V0,F = V0,B = V0 and inserting (10) in (8) and equating the coefficients of
equal powers of sn(αx, k) results in the following relations among the parameters
ωj , Cj , Aj and Bj and the characteristic of the optical lattice V0 , α and k
Nf
X
α2 k 2
Aj =
gBF
j=1
1
gBB
−
mB mF gBF
V0
gBB
−
1−
gBF
gBF
(19)
Nf
X
α2 (k 2 + 1)
α 2 k 2 B0
ω0 =
+ gBB B0 + gBF
Bi +
2mB
2mB A0
i=1
α 2 k 2 − mF V0
A0 =
,
mF gBF
α2 (k 2 + 1)
α 2 k 2 Bj
ωj =
+ gBF B0 +
2mF
2mF Aj
(20)
α 2 B0
α 2 Bj
(A0 +B0 )(A0 +B0 k 2 ),
Cj2 =
(Aj +Bj )(Aj +Bj k 2 ) (21)
A0
Aj
where j = 1, . . . , Nf . Next for convenience we introduce
C02 =
B0 = −β0 A0 ,
Bj = −βj Aj ,
j = 1, . . . , Nf
then
C02 = α2 A20 β0 (β0 − 1)(1 − β0 k 2 ),
Cj2 = α2 A2j βj (βj − 1)(1 − βj k 2 ).
228
Nikolay A. Kostov, Vladimir S. Gerdjikov and Tihomir I. Valchev
Table 1. Constraints ensuring the existence of type A solutions. Here
W = gBF mF WB /(mB WF ).
1
2
3
4
β0 ≤ 0
β0 ≤ 0
1 ≤ β0 ≤ 1/k 2
1 ≤ β0 ≤ 1/k 2
βj ≤ 0
1 ≤ βj ≤ 1/k 2
βj ≤ 0
1 ≤ βj ≤ 1/k 2
A0
A0
A0
A0
≥0
≥0
≤0
≤0
Aj
Aj
Aj
Aj
≥0
≤0
≥0
≤0
gBF
gBF
gBF
gBF
≷0
≷0
≷0
≷0
gBB
gBB
gBB
gBB
≶W
≷W
≷W
≶W
In order for our results (10) to be consistent with the parametrization (5)–(7) we
must ensure that both q0 (x) and Θ0 (x) are real-valued, and also qj (x) and Θj (x)
are real-valued; this means that C02 ≥ 0 and q02 (x) ≥ 0 and also Cj2 ≥ 0 and
qj2 (x) ≥ 0 (see Table 1, WB = (α2 k 2 − mB V0 ), WF = (α2 k 2 − mF V0 )). An
elementary analysis shows that with l = 0, . . . , Nf one of the following conditions
must hold
1
a) Al ≥ 0, βl ≤ 0
b) Al ≤ 0, 1 ≤ βl ≤ 2 ·
k
4. Type B Nontrivial Phase Solutions
For the first time solutions of this type were derived in [4, 5] for the case of nonlinear Schrödinger equation and in [7] for the n-component CNLSE. For Bose–Fermi
mixtures solutions of this type are possible
• when we have two lattices VB and VF .
• when mB = mF .
We seek the solutions in one of the following forms:
q02 = A0 sn(αx, k) + B0 ,
qj2 = Aj sn(αx, k) + Bj ,
q02
q02
qj2
qj2
= A0 cn(αx, k) + B0 ,
= A0 dn(αx, k) + B0 ,
j = 1, . . . , Nf (22)
= Aj cn(αx, k) + Bj
(23)
= Aj dn(αx, k) + Bj .
(24)
In the first case (22) we have
3α2 k 2
,
8mB
α2 k 2 Bj
A0 = −
,
4mF gBF Aj
VB =
X
Aj = −
j
ω0 =
α2 (k 2
+ 1)
8mB
VF =
3α2 k 2
8mF
BNf
B1
= ··· =
A1
AN f
α2 k 2 B0 A0 gBB
−
4mB gBF A0
gBF
+ gBB B0 + gBF B1 −
α2 k 2 B02
8mB A20
229
Bose-Fermi Mixtures in Two Optical Lattices
α2 (k 2 + 1)
α2 k 2 Bj2
+ gBF B0 −
8mF
8mF A2j
ωj =
C02 =
α2
(B 2 − A20 )(A20 − B02 k 2 ),
4A20 0
We remark that due to relations
B1
A1
Cj2 =
= ··· =
B Nf
A Nf
α2
(B 2 − A2j )(A2j − Bj2 k 2 ).
4A2j j
we have that all qj of the fermion
fields are proportional to q1 .
5. Examples of Elliptic Solutions
Using the general solution equations (11)–(13) we have the following special cases
(these solutions are possible only when we have some restrictions on gBB , gBF , and
V0 , see Table 1):
Example 1. Suppose that B0 = Bj = 0. Therefore we have
q0 (x) =
p
α2 k 2 − mF V0
,
A0 =
mF gBF
A0 sn(αx, k),
X
j
Aj =
qj =
q
Aj sn(αx, k)
α2 k 2
1
gBF
gBB
−
mB mF gFB
(25)
V0
gBB
−
1−
·
gBF
gBF
(26)
For the frequencies ω0 and ωj we have
ω0 =
α2 (1 + k 2 )
,
2mB
ωj =
α2 (1 + k 2 )
·
2mF
as well as C0 = Cj = 0.
Example 2. Let B0 = −A0 and Bj = −Aj hold true. Then we have
q0 (x) =
p
−A0 cn(αx, k),
qj (x) =
q
−Aj cn(αx, k).
(27)
The coefficients A0 and Aj have the same form as (26). The frequencies ω0 and
ωj now look as follows
α2 (1 − 2k 2 )
α2 (1 − 2k 2 )
+ V0 ,
ωj =
+ V0 .
2mB
2mF
The constants C0 and Cj are equal to zero again.
ω0 =
Example 3. B0 = −A0 /k 2 and Bj = −Aj /k 2 . In this case we obtain
√
p
−Aj
−A0
q0 (x) =
dn(αx, k),
qj (x) =
dn(αx, k)
k
k
α2 (k 2 − 2) V0
α2 (k 2 − 2) V0
ω0 =
+ 2,
ωj =
+ 2·
2mB
k
2mF
k
(28)
230
Nikolay A. Kostov, Vladimir S. Gerdjikov and Tihomir I. Valchev
As before C0 = Cj = 0.
Example 4. B0 = 0 and Bj = −Aj . The result reads
q
√
q0 (x) = A0 sn(αx, k),
qj (x) = −Aj cn(αx, k)
(29)
α2
α2 (1 − k 2 )
+ V0 + A0 gBB ,
ωj =
·
2mB
2mF
By analogy with the previous examples the constants A0 , Aj , C0 and Cj are given
by formulae (26) and C0 , Cj are all zero.
ω0 =
Example 5. B0 = 0 and Bj = −Aj /k 2 . Thus, one gets
q0 (x) =
p
p
−Aj
dn(αx, k)
k
α2 k 2
ωj =
·
2mF
A0 sn(αx, k),
qj (x) =
α2 (k 2 − 1) V0 A0 gBB
+ 2+
,
ω0 =
2mB
k
k2
Example 6. Let B0 = −A0 and Bj = 0. Hence we have
q0 (x) =
ω0 =
p
−A0 cn(αx, k),
qj (x) =
α2
− gBB A0 ,
2mB
ωj =
q
Aj sn(αx, k)
α2 (1 − k 2 )
+ V0 .
2mF
Example 7. Let B0 = −A0 and Bj = −Aj /k 2 . We obtain
q0 (x) =
ω0 =
p
p
−A0 cn(αx, k),
α2
1 − k2
V0
−
+
A0 gBB ,
k2
2mB
k2
−Aj
dn(αx, k)
k
α2 k 2
ω j = V0 −
·
2mF
qj (x) =
Example 8. Suppose B0 = −A0 /k 2 and Bj = 0. Then
√
q
−A0
q0 (x) =
dn(αx, k),
qj (x) = Aj sn(αx, k)
k
2
α k 2 gBB A0
α2 (k 2 − 1) V0
−
,
ω
=
+ 2·
ω0 =
j
2mB
k2
2mF
k
Example 9. Let B0 = −A0 /k 2 and Bj = −Aj . Thus
√
q
−A0
q0 (x) =
dn(αx, k),
qj (x) = −Aj cn(αx, k)
k
α2 k 2 k 2 − 1
V0
α2
ω0 = V0 −
+
g
A
,
ω
=
−
·
0
j
BB
2mB
k2
k2
2mF
All these cases when V0 = 0 and j = 2 are derived for the first time in [3].
(30)
231
Bose-Fermi Mixtures in Two Optical Lattices
Table 2. Constraints ensuring the existence of generic type B trivial
phase solutions. Here W = gBF mF WB /(mB WF ).
√
1 q0 = A0 sn(αx, k)
p
qj = Aj sn(αx, k)
√
2 q0 = −A0 cn(αx, k)
p
qj = −Aj cn(αx, k)
√
3 q0 = −A0 dn(αx, k)/k
p
qj = −Aj dn(αx, k)/k
√
4 q0 = A0 sn(αx, k)
p
qj = −Aj cn(αx, k)
√
5 q0 = A0 sn(αx, k)
p
qj = −Aj dn(αx, k)/k
√
6 q0 = −A0 cn(αx, k)
p
qj = Aj sn(αx, k)
√
7 q0 = −A0 cn(αx, k)
p
qj = −Aj dn(αx, k)/k
√
8 q0 = −A0 dn(αx, k)/k
p
qj = Aj sn(αx, k)
√
9 q0 = −A0 dn(αx, k)/k
p
qj = −Aj cn(αx, k)
gBF ≷ 0 gBB ≶ W
V0 ≶ α2 k 2 /mF
gBF ≷ 0 gBB ≶ W
V0 ≷ α2 k 2 /mF
gBF ≷ 0 gBB ≶ W
V0 ≷ α2 k 2 /mF
gBF ≷ 0 gBB ≷ W
V0 ≶ α2 k 2 /mF
gBF ≷ 0 gBB ≷ W
V0 ≶ α2 k 2 /mF
gBF ≷ 0 gBB ≷ W
V0 ≷ α2 k 2 /mF
gBF ≷ 0 gBB ≶ W
V0 ≷ α2 k 2 /mF
gBF ≷ 0 gBB ≷ W
V0 ≷ α2 k 2 /mF
gBF ≷ 0 gBB ≶ W
V0 ≷ α2 k 2 /mF
5.1. Mixed Trivial Phase Solution
Example 10. When
B0 = 0,
B1 = 0,
B2 = −A2 ,
Bj = −Aj /k 2 ,
j = 3, . . . , Nf .
the solutions obtain the form
q0 =
p
q2 =
p
A0 sn(αx, k),
−A2 cn(αx, k),
q1 =
p
qj =
q
A1 sn(αx, k)
−Aj dn(αx, k)/k.
Using equations (11)–(13) we have
Nf
α2 k 2 − V0 mF X
A0 =
,
Aj = α 2 k 2
mF gBF
j=1
ω0 =
1
gBB
−
2
mB gBF mF gBF
!
− V0
g A
α2 (k 2 − 1) gBF V0
BB 0
+ 2 A1 + (1 − k 2 )A2 +
+ 2
2mB
k
k2
k
1
gBB
− 2
gBF gBF
!
232
ω1 =
Nikolay A. Kostov, Vladimir S. Gerdjikov and Tihomir I. Valchev
α2 (1 + k 2 )
,
2mF
ω2 =
1
α2 ,
2mF
ωj =
α2 k 2
,
2mF
j = 3, . . . , NF .
Example 11. Let B0 = B1 = 0 and Bj = −Aj where j = 2, . . . , Nf . Therefore,
the solutions read
q0 (x) =
p
A0 sn(αx, k)
q1 (x) =
p
A1 sn(αx, k)
qj (x) =
q
−Aj cn(αx, k).
Then we obtain for frequencies the following results
ω0 =
α2 (1 − k 2 )
+ V0 + gBB A0 + gBF A1 ,
2mB
α2 (1 + k 2 )
,
2mF
ω1 =
ωj =
α2
·
2mF
Example 12. Suppose B0 = −A0 , B1 = 0, B2 = −A2 and Bj = −Aj /k 2 where
j = 3, . . . , Nf . The solutions have the form
q0 (x) =
p
q2 (x) =
p
−A0 cn(αx, k),
−A2 cn(αx, k),
q1 (x) =
p
qj (x) =
q
A1 sn(αx, k)
−Aj dn(αx, k)/k.
The frequencies are
V0
α2
1 − k2
gBF
−
+
(gBB A0 + gBF A2 ) + 2 A1
2
2
k
2mB
k
k
2
2
2
α (1 − k )
α (1 − 2k 2 )
ω1 = V 0 +
,
ω2 = V0 +
,
2mF
2mF
ω0 =
ωj = V0 −
α2 k 2
·
2mF
Example 13. Let B0 = −A0 , B1 = −A1 and Bj = −Aj /k 2 for j = 2, . . . , Nf .
Then
q0 (x) =
p
q1 (x) =
p
qj (x) =
q
−A0 cn(αx, k)
−A1 cn(αx, k)
−Aj dn(αx, k)/k
V0
α2
1 − k2
−
+
(gBB A0 + gBF A1 )
k2
2mB
k2
α2 (1 − 2k 2 )
α2 k 2
ω1 = V 0 +
,
ωj = V0 −
·
2mF
2mF
ω0 =
233
Bose-Fermi Mixtures in Two Optical Lattices
Example 14. Let B0 = −A0 /k 2 , B1 = −A1 and Bj = −Aj /k 2 for j =
2, . . . , Nf . Hence
q0 (x) =
p
−A0 dn(αx, k)/k
q1 (x) =
p
−A1 cn(αx, k)
qj (x) =
q
−Aj dn(αx, k)/k
α2 (k 2 − 2) V0 1 − k 2
+ 2+
(gBB A0 + gBF A1 )
2mB
k
k2
V0
α2
V0 α2 (k 2 − 2)
ω1 = 2 −
,
ωj = 2 +
·
k
2mF
k
2mF
ω0 =
Certainly these examples do not exhaust all possible combinations of solutions and
it is easy to it.
6. Vector Soliton Solutions
6.1. Vector Bright-Bright Soliton Solutions
When k → 1, sn(αx, 1) = tanh(αx) and B0 = −A0 , Bj = −Aj we obtain that
the solutions read
q
p
1
1
q0 = −A0
,
qj = −Aj
cosh(αx)
cosh(αx)
where A0 ≤ 0 as well as Aj ≤ 0. Using equations (11)–(13) we have
A0 =
Nf
X
α2 − V0 mF
,
mF gBF
α2
Aj =
gBF
j=1
ω0 = V0 −
V = V0 tanh2 (αx)
1
gBB
−
mB mF gBF
1
α2 ,
2mB
V0
gBB
−
1−
gBF
gBF
ωj = V0 −
1
α2 .
2mF
As a consequence of the restrictions on A0 and Aj one can get the following unequalities
gBF > 0,
gBF < 0,
α2
,
mF
α2
V0 ≤
,
mF
V0 ≥
(α2 − mB V0 )mF
gBF
(α2 − mF V0 )mB
(α2 − mB V0 )mF
≥ 2
gBF .
(α − mF V0 )mB
gBB ≤
gBB
Vector bright soliton solution when V0 = 0 is derived for the first time in [3].
234
Nikolay A. Kostov, Vladimir S. Gerdjikov and Tihomir I. Valchev
6.2. Vector Dark-Dark Soliton Solutions
When k → 1 and B0 = Bj = 0 are satisfied the solutions read
q0 (x) =
p
A0 tanh(αx),
qj (x) =
q
Aj tanh(αx).
The natural restrictions A0 ≥ 0 and Aj ≥ 0 lead to
gBF > 0,
gBF < 0,
A0 =
α2 − mF V0
,
mF gBF
(α2 − mB V0 )mF
gBF ,
V0 ≤ α2 /mF
(α2 − mF V0 )mB
(31)
(α2 − mB V0 )mF
2
gBB ≥ 2
gBF ,
V0 ≥ α /mF
(α − mF V0 )mB
X
α2
1
gBB
V0
gBB
Aj =
−
−
1−
·
gBF mB mF gFB
gBF
gBF
j
gBB ≤
For the frequencies ω0 and ωj and the constants C0 and Cj we have
ω0 =
α2
,
mB
ωj =
α2
,
mF
C0 = Cj = 0.
(32)
6.3. Vector Bright-Dark Soliton Solutions
When k → 1, B0 = −A0 and Bj = 0, we have
√
q
−A0
q0 (x) =
,
qj (x) = Aj tanh(αx)
cosh(αx)
α2
ω0 =
− gBB A0 ,
ωj = V0 ,
C0 = Cj = 0.
2mB
The parameters A0 and Aj are given by (31). In this case we have the following
restrictions
(α2 − mB V0 )mF
gBF ,
(α2 − mF V0 )mB
(α2 − mB V0 )mF
gBF ,
≤ 2
(α − mF V0 )mB
gBF > 0,
gBB ≥
V0 ≥ α2 /mF
gBF < 0,
gBB
V0 ≤ α2 /mF .
6.4. Vector Dark-Bright Soliton Solutions
When k → 1 and provided that B0 = 0 and Bj = −Aj the result is
√
q0 (x) =
−Aj
α2
, ω0 = V0 + A0 gBB , ωj =
·
cosh(αx)
2mF
p
A0 tanh(αx), qj (x) =
235
Bose-Fermi Mixtures in Two Optical Lattices
By analogy with the previous examples the constants A0 , Aj , C0 and Cj are given
by formulae (31) and (32), respectively. The restrictions now are
(α2 − mB V0 )mF
gBF ,
(α2 − mF V0 )mB
(α2 − mB V0 )mF
≤ 2
gBF ,
(α − mF V0 )mB
gBF > 0,
gBB ≥
V0 ≤ α2 /mF
gBF < 0,
gBB
V0 ≥ α2 /mF .
6.5. Vector Dark-Dark-Bright Soliton Solutions
Let B0 = B1 = 0 and Bj = −Aj where j = 2, . . . , Nf . Therefore the solutions
read
q0 (x) =
p
A0 tanh(αx), q1 (x) =
p
A1 tanh(αx), qj (x) =
q
−Aj sech(αx).
Then we obtain for frequencies the following results
ω0 = V0 + gBB A0 + gBF A1 ,
ω1 =
α2
,
mF
ωj =
α2
·
2mF
These examples are by no means exhaustive.
6.6. Nontrivial Phase, Trigonometric Limit
In this section we consider a trap potential of the form Vtrap = V0 cos(2αx), as
a model for an optical lattice. Our potential V is similar and differs only with
additive constant. When k → 0, sn(αx, 0) = sin(αx)
q02 = A0 sin2 (αx) + B0 ,
qj2 = Aj sin2 (αx) + Bj
(33)
1
V = V0 sin2 (αx) = (V0 − V0 cos(2αx)).
(34)
2
Using equations (11)–(13) again we obtain the following result when (see Table 3)
Nf
X
V0
gBB
1−
Aj = −
gBF
gBF
j=1
V0
A0 = −
,
gBF
Table 3. W = gBF mF WB /(mB WF ).
1
2
3
4
β0
β0
β0
β0
≤0
≤0
≥1
≥1
βj
βj
βj
βj
≤0
≥1
≤0
≥1
A0
A0
A0
A0
≥0
≥0
≤0
≤0
Aj
Aj
Aj
Aj
≥0
≤0
≥0
≤0
gBF
gBF
gBF
gBF
≷0
≷0
≷0
≷0
gBB
gBB
gBB
gBB
≶ gBF
≷ gBF
≷ gBF
≶ gBF
V0
V0
V0
V0
≶0
≶0
≷0
≷0
236
Nikolay A. Kostov, Vladimir S. Gerdjikov and Tihomir I. Valchev
Nf
X
1 2
Bi ,
α + B0 gBB + gBF
ω0 =
2mB
i=1
C02 = α2 B0 (A0 + B0 ),
ωj =
1 2
α + gBF B0
2mF
Cj2 = α2 Bj (Aj + Bj )
where
s
A0 + B0
tan(αx)
B0
s
Aj + Bj
tan(αx) .
Bj
Θ0 (x) = arctan
Θj (x) = arctan
!
!
This solution is the most important from the physical point of view [16].
7. Linear Stability, Preliminary Results
To analyze linear stability of our initial system of equations we seek solutions in
the form
iω0
ψ0 (x, t) = (q0 (x) + εφ0 (x, t)) exp −
t + iΘ0 (x) + iκ0
~
iωj
ψj (x, t) = (q1 (x) + εφj (x, t)) exp −
t + iΘ1 (x) + iκ1
~
and obtain the following linearized equations





~
Φ0
Φ1
..
.





ΦNf
,t
Λ0 U1 U2
V
 1 Λ1 0

V
0 Λ2
=
 .2
..
..
 .
 .
.
.
VNf 0 0
Φ0 =
φR
0

. . . UNf 

Φ0
... 0 
 Φ 

1 
... 0 
 . 

..   .. 
..
.
. 
ΦNf
. . . ΛNf
!
,
φI0 ,
Φj =
φR
j
!
φIj
where
Λ0 =
S0 L0,−
,
L0,+ S0
Uj =
0 0
U0,j 0
and
0 0
Vj =
U1,j 0
C0
1
S0 = −
∂x
,
mB q0
q0
Cj
Sj = −
∂x
mF qj
Sj Lj,−
Λj =
,
Lj,+ Sj
1
qj
!
237
Bose-Fermi Mixtures in Two Optical Lattices
L0,−
1
=−
2mB
L0,+
1
=
2mB
Lj,−
1
=−
2mF
Lj,+
1
=
2mF
2
∂xx
2
∂xx
C2
− 04
q0
2
∂xx
2
∂xx
C2
− 04
q0
+ V + gBB q02 + gBF q12 − ω0
!
− V − 3gBB q02 − gBF q12 + ω0
Cj2
− 4
q0
Cj2
− 4
q0
!
!
+ V + gBF q02 − ωj
!
− V − gBF q02 + ωj
U0,j = −2gBF q02 ,
U1,j = −2gBF q0 qj .
The analysis of the latter matrix system is a difficult problem and only numerical
simulations are possible. Recently a great progress was achieved for analysis of
linear stability of periodic solutions of type (5), (6) (see, e.g., [4, 5, 7, 11] and
references therein). Nevertheless the stability analysis is known only for solutions
of type (25)–(30) and solutions with nontrivial phase of type (33) and (34). Linear
analysis of soliton solutions is well developed, but it is out scope of the present
paper.
Finally we discuss three special cases:
Case p
I. Let B0 = Bj = 0 then for j = 1, . . . , Nf and q0 =
qj = Aj sn(αx, k) we have the following linearized equations

√
A0 sn(αx, k),

~φR
0,t
X
1 2 I 
= −
∂xx φ0 + V0 + gBB A0 + gBF
Aj  sn2 (αx, k)φI0 − ω0 φI0
2mB
j
~φI0,t
X
1 2 R 
=
∂xx φ0 − V0 + 3gBB A0 + gBF
Aj  sn2 (αx, k)φR
0
2mB
j

2
+ ω0 φ R
0 − 2gBF A0 sn (αx, k)

X
φR
j
j
~φR
j,t
~φIj,t
1 2 I
∂ φ + (V0 + gBF A0 ) sn2 (αx, k)φIj − ωj φIj
= −
2mF xx j
1 2 R
R
=
∂ φ − (V0 + gBF A0 ) sn2 (αx, k)φR
j + ωj φ
2mF xx j
q
− 2gBF A0 Aj sn2 (αx, k)φR
0.
238
Nikolay A. Kostov, Vladimir S. Gerdjikov and Tihomir I. Valchev
√
Case II. Let B0 = −A0 , Bj = −Aj then for q0 = −A0 cn(αx, k), qj =
−Aj cn(αx, k) we obtain the following linearized equations
p


~φR
0,t
X
1 2 I 
∂xx φ0 + V0 + gBB A0 + gBF
Aj  sn2 (αx, k)φI0
= −
2mB
j


− gBB A0 + gBF
X
Aj + ω0  φI0
j


~φI0,t
X
1 2 R 
=
∂xx φ0 + 3gBB A0 + gBF
Aj + ω0  φR
0
2mB
j


− V0 + 3gBB A0 + gBF
X
Aj  sn2 (αx, k)φR
0
j
2
+ 2gBF A0 (1 − sn (αx, k))
X
φR
j
j
~φR
j,t
~φIj,t
1 2 I
= −
∂ φ + (V0 + gBF A0 ) sn2 (αx, k)φIj − (gBF A0 + ωj )φIj
2mF xx j
1 2 R
R
∂ φ − (V0 + gBF A0 ) sn2 (αx, k)φR
=
j + (gBF A0 + ωj )φj
2mF xx j
q
− 2gBF A0 Aj (1 − sn2 (αx, k))φR
0,
j = 1, . . . , Nf .
Case III. Let B0 = −A0 /k 2 , Bj = −Aj /k 2 therefore the solutions are
q0 =
p
−A0 dn(αx, k)/k,
qj =
q
−Aj dn(αx, k)/k
and we obtain the following linearized equations

~φR
0,t

X
1 2 I 
= −
∂xx φ0 + V0 + gBB A0 + gBF
Aj  sn2 (αx, k)φI0
2mB
j


− gBB A0 + gBF
X
A j + k 2 ω0 
j

~φI0,t
φI0
k2

X
1 2 R 
φR
=
∂xx φ0 + 3gBB A0 + gBF
Aj + k 2 ω0  02
2mB
k
j

− V0 + 3gBB A0 + gBF

X
j
Aj  sn2 (αx, k)φR
0
Bose-Fermi Mixtures in Two Optical Lattices
+
239
2gBF A0 (1 − k 2 sn2 (α, k)) X R
φj
k2
j
1 2 I
∂ φ + (V0 + gBF A0 ) sn2 (αx, k)φIj
2mF xx j
gBF A0 + k 2 ωj I
−
φj
k2
1 2 R
gBF A0 + k 2 ωj R
~φIj,t =
∂xx φj − (V0 + gBF A0 ) sn2 (αx, k)φR
φj
j +
2mF
k2
p
2gBF A0 Aj (1 − k 2 sn2 (α, k))φR
0
,
j = 1, . . . , Nf .
−
k2
These cases are by no means exhaustive.
~φR
j,t = −
8. Conclusions
In conclusion, we have considered the mean field model for boson-fermion mixtures in two optical lattices. Classes of quasi-periodic, periodic, elliptic solutions
have been analyzed. These solutions can be used as initial states which can generate localized matter waves (solitons) through the modulational instability mechanism. This important problem is under consideration.
Acknowledgements
We thank the Bulgarian Science Foundation for a partial support under the contract
# F-1410.
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