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Quantum breathers and intrinsic localized excitation associated with the modulational instability in 1D Bose–Hubbard chain
Accepted Manuscript
Quantum breathers and intrinsic localized excitation associated with the Modulational instability in 1D Bose-Hubbard chain Z.I. Djoufack, E. Tala-Tebue, J-P Nguenang PII: DOI: Reference:
S1007-5704(18)30231-4 10.1016/j.cnsns.2018.07.018 CNSNS 4591
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
10 June 2017 15 June 2018 9 July 2018
Please cite this article as: Z.I. Djoufack, E. Tala-Tebue, J-P Nguenang, Quantum breathers and intrinsic localized excitation associated with the Modulational instability in 1D Bose-Hubbard chain, Communications in Nonlinear Science and Numerical Simulation (2018), doi: 10.1016/j.cnsns.2018.07.018
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• In this work, the energy spectrum and intrinsic localized modes associated with the modulational instability of boson chains can be described by a Bose-Hubbard model. By using of number states method combined with the numerical diagonalization, we show that the energy spectrum of the system exhibits bound states signature. With the help of multiple scales method in addition to a quasidiscreteness approximation, we obtain bright and dark type intrinsic localized modes at the center and at the edges of the Brillouin zone respectively and their appearance conditions. We found that the inclusion of interaction of two bosons at the same site can influence the appearance conditions of bright and dark solitons solution.Considering the fact that nowadays, the forthright way to predict the formation of intrinsic localized modes in nonlinear systems is the modulational instability. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in the Bose-Hubbard chain. It is found that the shape of the region of modulational instability and the instability growth rates of the plane wave can be more affected dramatically when the interaction of two bosons at the same site is involved in the system. Furthermore, we calculate the instability growth rates of plane wave at the center and at the edge of the Brillouin zone for different value of the interaction of two bosons to analyze their formation conditions which are in full agreement with the method of multiple scale in addition to a quasidiscreteness approximation analysis
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Highlights
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Quantum breathers and intrinsic localized excitation associated with the Modulational instability in 1D Bose-Hubbard chain Z. I. Djoufack
1,2 ∗
, E. Tala-Tebue 1 , and J-P Nguenang
3
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1 Fotso Victor University Institute of Technology, Department of Telecommunication and Network Engineering, P.O. Box 134 Bandjoun, University of Dschang, Cameroon 2 African Institute for Mathematical Sciences, 6 Melrose, Muizenberg, Cape Town 7945, South Africa and 3 Fundamental physics laboratory: Group of Nonlinear physics and Complex systems, Department of Physics, University of Douala, P.O. Box 24157, Douala, Cameroon (Dated: July 10, 2018)
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In this work, the energy spectrum and intrinsic localized modes associated with the modulational instability of boson chains is described by a Bose-Hubbard model. By using of number states method combined with the numerical diagonalization, we show that the energy spectrum of the system exhibits bound states signature. With the help of multiple scales method in addition to a quasidiscreteness approximation, we obtain bright and dark type intrinsic localized modes at the center and at the edges of the Brillouin zone respectively and their appearance conditions. We found that the inclusion of interaction of two bosons at the same site can influence the appearance conditions of bright and dark soliton solutions. Considering the fact that nowadays, the forthright way to predict the formation of intrinsic localized modes in nonlinear systems is the modulational instability. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in the Bose-Hubbard chain. It is found that the shape of the region of modulational instability and the instability growth rates of the plane wave can be more affected dramatically when the interaction of two bosons at the same site is involved in the system. Furthermore, we calculate the instability growth rates of plane wave at the center and at the edge of the Brillouin zone for different values of the interaction of two bosons to analyze their formation conditions which are in full agreement with the method of multiple scale in addition to a quasi-discreteness approximation analysis.
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Keywords: Quantum breathers; Bose-Hubbard model; Modulational instability and intrinsic localized modes.
INTRODUCTION
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I.
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PACS numbers: 05.45.Yv, 52.35.Sb, 42.65.Tg, 42.81.Dp
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Localized nonlinear excitation, generic time periodic and spatially localized solutions of the nonlinear systems with translational invariance due to the interplay between nonlinearity and discreteness are called discrete breathers [1, 2]. The existence of breather excitations in 1D lattices was first established by MacKay and Aubry [3]. Discrete breathers which play important roles in both energy storage and transport in various low dimension materials [4] have attracted much attention of intense theoretical and experimental subjects [5–10]. Experimentally, Discrete breathers have been observed in many physical systems such lattice vibrations in crystals, magnetic solids, Josephson junction, photonic crystals and antiferromagnetic structures [2], to mention a few. These latter studies demonstrate that intensive works have been investigated to master discrete breathers in classical nonlinear lattices. However, their quan-
∗ Corresponding
author: [email protected] (Zacharie Isidore Djoufack)
tum counterparts namely quantum breathers are less known nowadays and attract the attention of searchers in different domains such as mathematical and physical systems. In quantum domain, quantum breathers consist of superpositions of nearly degenerate many quanta bound states, characterized with long times to tunnel from one lattice site to another [11]. In the past, by means of numerical exact diagonalization, Wang et al. found the existence of quantum breathers in Klein Gordon atomic lattice model [12]. Furthermore, several works have been investigated on quantum properties of phonons [13–18]. Very recently, we found that two, four, and six-quanta quantum breathers exist in ferromagnetic spin chains using perturbation theory and exact numerical diagonalization [19, 20]. Using the similar technique namely, perturbation theory and numerical diagonalization, some works have shown the existence of quantum breathers in Bose-Hubbard model [21–23] and in 1D ultracold bosons in optical lattices [24]. A basic process in the theory of nonlinear wave is the modulational instability [25–29]. Modulational in-
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THE MODEL
The Hamiltonian of Bose-Hubbard discrete with bosonic operators is read: H1 = −
f h X j=1
i γa†j a†j aj aj + 1 (a†j aj+1 + aj a†j+1 ) , (1)
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where γ is the parameter controlling the strength of the interaction in the lattice and well known as an interaction of two bosons at the same site in the lattice, a†j and aj are the bosonic creation and annihilation operators. These operators satisfy the following commutation relations: [ai , a†j ] = δij , [a†i , a†j ] = [ai , aj ] = 0. While rescaling (1) by 1 , it can be written as f h i X f1 = H1 = − H ηa†j a†j aj aj + a†j aj+1 + aj a†j+1 ,(2) 1 j=1
where η = γ/1 . Due to the fact that the chain is tanslationally invariant, the Hamiltonian of this quantum system commutes with the number operator b = Pf a† aj , whose eigenvalue is n. N j=1 j
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stability is considered as an universal phenomenon that occurs in many domains such as optics, hydrodynamics, biology, to mention a few [30–35, 37, 38]. Modulational instability is defined as dynamical instabilities characterized by an exponential growth of amplitude perturbations which can occur during propagation under the interplay between nonlinearity and dispersion. Modulational instability in nonlinear lattice is the best way to predict the formation intrinsic localized modes [39, 40]. It is considered to some extent, a precursor to soliton formation. Kivshar and Peyrard were the first to establish that the modulational instability can be used to explore the generation of localized states in nonlinear lattices. They also showed that the discreteness can drastically modify the conditions for modulational instability [41]. Moreover, it was shown by Kivshar [42] that when the system displays modulational instability, only bright intrinsic localized mode exist. Daumont et al. predicted the evolution of a linear wave in the presence of noise, and also proved that modulational instability is the first step towards energy localization [43].
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However, in the case of the Bose-Hubbard model very few studies have been mostly devoted to analyze the non-equilibrium quantum dynamics of a Bose-Einstein condensate and density modulations associated with the dynamical instability [44, 45]. From the foregoing, it is clear that for better understanding of the generation of the bright soliton and the intrinsic localized mode in Bose-Hubbard chains, it is normal to improve investigation in the modulational instability. The Bose-Hubbard model is a very interesting nonlinear lattice which can help to better understand quantitatively the role played by the lattice interaction of two bosons at the same site and attracts our attention.
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The purpose of this paper is to explore quantum breathers and intrinsic localized excitation associated with the modulational instability in 1D Bose-Hubbard chains to analyze the influence of the interaction of two bosons at the same site on the properties of quantum breathers, intrinsic localized excitation and modulational instability in this nonlinear system. This paper is organized as follows: the BoseHubbard model is described in section II. In section III, we use the number state method combined with numerical diagonaliztion to prove that bound states can be controlled by the interaction between two bosons. In section IV, to predict the formation conditions of intrinsic localized modes, a linear analysis is carried out by means of the multiple scale method in addition to a quasi-discreteness approximation. Modulational instability/stability criterion and growth rates of plane waves are investigated in section V. Section VI concludes the paper.
III. INTERACTION BETWEEN TWO BOSONS IN THE BOSE-HUBBARD CHAIN
To study the two bosons’ case, we use a position state representation to describe the quantum states, | ψj =| n1 , n2 , · · · , nf , where P nj stands for the number of bosons at site j (n = j nj ). For instance, the state | 20100 · · · 0 represents a state with 2 bosons at the site 1, another boson at site 3 and nothing elsewhere. One can apply the translation operator to these states. This can be explained by the fact that chain is subjected to periodic boundary conditions. For a given number of bosons, each eigenstate is a linear combination of the number state with fixed n. For the sake of simplicity, we consider in this work an odd number of sites f . A general eigenfunction of these states for the Hamiltonian (2) is a Bloch wave. In the case for two bosons, this eigenfunction is read:
|ψ =
(f +1) 2
X j=1
f X Tb s−1 1 Cj | ψj = √ C1 | 20 · · · 0 τ f s=1
+C2
f X Tb s−1 s=1
+ · · · + C (f +1) 2
τ
f X Tb s−1 s=1
τ
| 110 · · · 0 (3)
| 10 · · · 01 ,
here Tb is the translation operator and k = 2πν/f , f −1 ik with ν ∈ {− f +1 represents the 2 , 2 } and τ = e eigenvalue of the translational operator Tb. To verify
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f +1 2
+τ
f +1 2
).
ENERGY SPECTRUM
0 -5
-10 -1
10
0
0.5
1
0
0.5
1
-0.5
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The eigenspectra of (4) plotted in Fig. 1 for different values of the parameter η is obtained by using the numerical diagonalization method. As we can see in Fig. 1(a), in this attractive case (η > 0), the isolated bound state is localized below the continuum band. From a physical picture applied to this quantum lattice, the isolated bound state represents the case where two bosons occupy the same site along the lattice and the probability of finding two bosons on the same site is high, confirming that the isolated bound state is a localized quantum sate (see Fig. 2). In the absence of the interaction of two bosons at the same site (η = 0), the energy spectrum in this case is only composed by the continuum band. Figure 1(b) shows the repulsive case for η < 0 with the black curve on top.
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where q1 = 1+τ 1 , q1∗ = 1+τ −1 and p = τ −
(a)
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that ψ | ψ = 1, it is important that C1 , C2 , ... must P (f +1) 2 | Cj |2 = 1. We can generbe normalized as j=1 ate an equivalence class of states with the aid of the translation operator Tb defined as Tba†j = a†j+1 Tb, so that Tb[n1 , n2 , · · · , nf ] = [nf , n1 , · · · , nf −1 ]. With the help of the basis given in (3), it is obvious to construct the following Hamiltonian matrix √ ∗ 2q1 0 0 0 0 0 0 √η 2q1 0 q1∗ 0 0 0 0 0 0 q1 0 0 0 0 0 0 .. .. .. 0 . . . 0 0 0 0 f1 = − H (4) . . . . . . . . . . 0 . . . . . 0 0 .. .. .. .. 0 . . . . q1∗ 0 0 0 0 0 0 0 q1 0 q1∗ 0 0 0 0 0 0 q1 p
5
0
-5 -1
-0.5
k/π
FIG. 1: (color online) Energy spectrum for two bosons’case for different values of η, f = 101: (a) η = 10 for attractive case; (b) η = −10 for repulsive case. 1
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LINEAR STABILITY ANALYSIS 0.8
aj |Φj i = Φj |Φj i,
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a†j |Φj i = Φ∗j |Φj i, ∞ 1 X (Φj )l √ |li. |Φj i = exp − |Φj |2 2 l! l=1
(5)
Φj in (5), represents the coherent amplitude of the operator aj for the system in the state |Φj i. The system state |Ψ(t)i in the coherent state representation, is defined by the product of the multimodes coherent state which can take the following form Y Φj , Ψ(t) = (6) j
(6) should be normalized as hΨ(t)|Ψ(t)i = 1. With the aid of Ehrenfest’s theorem [47], the dynamics of
Cj
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To describe the components of quantum state of the Bose-Hubbard chains, we adopt the Glauber’s coherent-state [46] defined as:
2
IV.
k/π (b)
0.6
0.4
0.2
0 0
20
40
60
Sites j
80
100
120
FIG. 2: square wave amplitudes of the eigenvectors in real space. Here η = 10, f = 101
bosons can be expressed in terms of the Heisenberg equation of motion for the Bose operator annihilation aj as i
dhaj i f1 ]i. = h[aj , H dt
(7)
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dΦj = −(Φj+1 + Φj−1 ) − η|Φj |2 Φj . dt
where A is an envelope function of variables τ and ζj which will be determined later. Inserting (16) into (15), one can get the following equation which stands for the boson linear dispersion relation ω(k) = −2 cos(ka).
(8)
Φj (t) = Φ
(1)
2
(τ, ζj , αj ) + Φ
(2)
(τ, ζj , αj ) ∞ X +··· = µ Φµj,j ,
(9)
µ=1
i
∂ (2) (2) (2) Φ + Φj,j−1 + Φj,j+1 ∂t j,j h i ∂ = i λ − Vg A(τ, ζj ) exp(iαj ), ∂ζj i
∂ (m) (m) (m) (m) Φ + Φj,j−1 + Φj,j+1 = γj,j , (m = 1, 2, 3 · · · ) ∂t j,j (10)
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with (1)
γj,j = 0,
λ=
dω = Vg = 2a sin(ka), dk
(19)
it is important to note here that λ is the group velocity. By solving (18) and considering the condition (19), (18) admits the special solution in the form (2)
Φj,j = B(τ, ζj ) exp(iαj ),
(20)
where B = B(τ, ζj ) is another function which will be determined in higher-order approximations. Next, if we set m = 3 in (10), considering the results previously obtained in (16), (17), (19) and (20) respectively, then we get ∂ (3) (3) (3) Φ + Φj,j−1 + Φj,j+1 ∂t j,j h ∂A i ∂2A =− i + a2 cos(ka) 2 + η|A|2 A exp(iαj ). ∂τ ∂ζj i
∂ (2) ∂ (1) a2 ∂ 2 (1) (1) Φj,j − i Φj,j − Φ + Φ j,j−1 j,j+1 ∂ζj ∂τ 2 ∂ζj2 (21) ∂ (1) (1) (1) (1) +a Φj,j−1 − Φj,j+1 − η|Φj,j |2 Φj,j , ∂ζj (13) In order to valid the theory, one must ensure that the secular term in the right hand side of (21) disap(4) pears. Therefore, one can obtain the following evoluγj,j = · · · . (14) tion form of nonlinear amplitude A(τ, ζj ) equation
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(3)
γj,j = iλ
∂ (1) ∂ (1) (1) Φj,j−1 − Φj,j+1 , (12) Φj,j + a ∂ζj ∂ζj
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(2)
γj,j = iλ
(11)
(18)
where Vg = dω/dk is the group velocity of the carrier waves. It should be noticed here that the term proportional to exp(iαj ) on the right hand side of Eq.(18) is a secular term which should be canceled in order to valid the theory [50]. Therefore, we get
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where is a small finite parameter whose represents the relative amplitude of the excitations, Φµj,j ≡ Φ(µ) (τ, ζj , αj ), ζj are slow variables defined by ζj = (ja − λt) and τ = 2 t respectively. αj = kja − ωt is the ”fast” variable, stands for the phase of the carrier wave number, a is lattice constant, k is the wave number and ω represents the frequency of the carrier wave. Inserting expansion (9) into (8), we obtain the following equations:
If we set m to 2 (m = 2) in (10), then we can get the second-order approximation equation given as
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Due to its nonlinearity and discreteness, Eq.(8) is not easy to solve exactly. This is why in this work, we adopt the method of multiple scales combined with a quasi-discreteness approximation [48, 49] to find solution of (8). In the treatment, we first set
(17)
m For the sake of simplicity, the expressions of γj,j (m = 4, 5, · · · ) are not written down completely here. For the lowest order of , which means the case where m = 1, one can obtain the following linear wave equation
i Taking
∂ (1) (1) (1) Φ + Φj,j−1 + Φj,j+1 = 0. ∂t j,j
into
account
the
fact
that
(15) (1)
Φj,j
=
(1) Φj,j (τ, ζj , αj ),
we get the general solution of (15) written in the following form (1)
Φj,j = A(τ, ζj ) exp(iαj ), αj = kja − ωt,
(16)
i
∂A ∂2A + P 2 + Q|A|2 A = 0, ∂τ ∂ζj
(22)
where P = a2 cos(ka) and Q = η. With the help of the following transformation A(τ, ζj ) = (1/ε)u(Zj , t), ζj = ε(ja − Vg ) = εZj and considering that τ = ε2 t, (22) be rewritten in the following form i
∂u ∂2u +P + Q|u|2 u = 0, ∂t ∂Zj2
(23)
where Zj = ja − Vg . Eq.(23) is the nonlinear Schr¨odinger equation (NLS). Depending on the sign
ACCEPTED MANUSCRIPT 6 whereas we employ the method of multiple scale combined with a quasi-discreteness approximation. In Heisenberg antiferromagnetic chains [52], analogous excitations have been found. Taking into account that there is a direct link between the formation of the bright type intrinsic localized modes and modulational instability of plane waves thus, we predict the appearance of modulationnal instability of plane wave at the center of the brillouin zone in presence of the interaction of two bosons at the same site when it takes positive value corresponding to an attractive case.
1. First case: k=0. In this case, we have
On the other hand, we have seen that in this first case that, due to its intrinsic nonlinearity, the Bose-Hubbard chains may support bright soliton solution. From physical picture, it is well known that the bright envelope soliton can propagate without losing energy. This implies that the bright soliton in Bose-Hubbard chains may be important in quantum system such as in quantum information.
d2 ω > 0, dk 2 ω = ωmin = −2, 1 00 P1 = P (k = 0) = ωmin = a2 , 2 Q1 = Q(k = 0) = η. Vg = 0,
(24) (25) (26)
For η < 0, we find that Q1 < 0 and the sign of the product P Q < 0 and (23) has the dark soliton solution given by
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(27)
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of product P Q, the solution of this equation can be localized bright soliton solution which has a vanishing amplitude at Zj 7−→ ∞ if the sign of product P Q > 0, i.e., P and Q have the same sign. The NLS equation has a dark soliton solution where the sign of the product P Q < 0. These configurations are discussed in this study to specify different types of intrinsic localized wave modes in nonlinear lattice model. The group velocity of the traveling wave is equal to zero at the center of the Brillouin zone for k = 0 and at the edges of the Brillouin zone for k = ±(π/a)
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P1 > 0 while the sign of Q1 depend on the value of η. We find that for η > 0, then Q1 > 0 and the sign of the product P Q is positive and (23) has the bright soliton solution s 00 h i ωmin A0 sech A0 (j − j0 )a (28) u= Q1 1 00 × exp[i ωmin A20 t + ϕ0 )], 2
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where A0 and ϕ0 are integral constants. j0 is an arbitrary integer. Thus we have
(29)
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Φj (t) = Φ1j,j (t) = u(Zj , t) exp(−iωmin t) s 00 h i ωmin = A0 sech A0 (j − j0 )a Q1 with
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× exp(−iΩ1 t + ϕ0 ),
Ω1 = ωmin − a2 A20 .
(30)
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Eq.(29) is the bright soliton solution which localized at the lattice site j = j0 and whose eigenfrequency Ω1 < ωmin is below the bottom of the linear wave spectrum. The site j = j0 depends on the initial exciting condition of the Bose-Hubbard model, is the central position of the localized mode. This means that the system is translational symmetry and we conclude that such intrinsic localized modes may be excited at any lattice site in the Bose-Hubbard model. The time evolution of the square of the coherent-state amplitude for this bright envelope soliton obtained at the center of the Brillouin zone is shown in Fig. 3 (a) The bright type intrinsic localized modes obtained in this first case is analogous to the one found in classical ferromagnetic chains with on-site anisotropy [51] where authors used the continuum approximation
u(Zj , t) = A(Zj , t) exp(iϕ(Zj , t)),
(31)
where A(Zj , t) is the amplitude function which is given as follows n hp io 12 A(Zj , t) = A0 1 − σ 2 sech C1 σA0 (j − j0 )a , (32)
with σ[0, 1], and stands for a parameter controlling the depth of the modulation of the amplitude function whereas C1 = −Q1 /2P1 = −η/2a2 . The expression of the phase function ϕ(Zj , t) is p n p ϕ(Zj , t) = C1 A0 1 − σ 2 (j − j0 )a h p io σ + tan−1 √ tanh C1 A0 σ(j − j0 )a 1 − σ2 Q1 − (3 − σ 2 )A20 t. (33) 2 Inserting Eqs.(32) and (33) into (31), then we have the dark soliton solution in the following form Φj (t) = u(Zj , t) exp(−iωmin t) n hp io 21 = A0 1 − σ 2 sech C1 σA0 (j − j0 )a hp n p × exp i C1 A0 1 − σ 2 (j − j0 )a h p io σ + tan−1 √ tanh C1 A0 σ(j − j0 )a 1 − σ2 i −iΩ2 t , (34)
where Ω2 = ωmin − η2 (3 − σ 2 )A20 . Eq.(34) stands for a dark type intrinsic localized mode in nonlinear lattice
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(a)
ond case, we get d2 ω < 0, dk 2 ω = ωmax = 2, 1 00 P2 = P (k = ±π/a) = ωmax = −a2 , 2 Q2 = Q(k = ±π/a) = η. Vg = 0,
0.1 0 2
50
1 Time
0 −50
0 Sites j
(b) u=
s
00 h i |ωmax | A0 sech A0 (j − j0 )a Q2 1 00 × exp[i ωmax A20 t + ϕ0 )], 2
(36) (37) (38)
(39)
thus it is not difficult to get
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Φj
2
0.2 0.1 0 2 0 −50
0 Sites j
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FIG. 3: Time evolution of the intensity of the coherentstate amplitude for the bright and dark soliton in the center of the Brillouin zone, where a = 1, f = 95 and σ = 0.5: a) k = 0 and η = 51.2; b) k = π/a and η = −39.8
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k = ±(π/a).
(40)
× exp(−iΩ3 t + ϕ0 ),
with
Ω3 = ωmax + a2 A20 .
(41)
Eq. (40) is the bright soliton solution whose eigenfrequency Ω3 > ωmax is below the upper of the linear wave spectrum. For η > 0, Q2 > 0 and the sign of the product P Q < 0 and (23) has the dark soliton solution given by (31) where the amplitude function is given as follows
models, whose eigenfrequency Ω2 lies the harmonic wave frequency band above. Then, we note here that this localized mode is a dark intrinsic localized nonlinear resonant mode in Bose-Hubbard system. The time evolution of the square of the coherent-state amplitude for this dark type intrinsic localized mode is shown in Fig. 3 (b) for a wave number k = 0 at the center of the Brillouin zone. As we can see in Fig. 3 (b), the time evolution of this nonlinear plane wave is symmetry and stable, thus such intrinsic localized modes can be excited at any lattice site in the Bose-Hubbard chains. We can predict here that the modulational instability will not take place in the system at the center of the Brillouin zone when the interaction of two bosons at the same site takes negative value corresponding to a repulsive case. Second case:
Φj (t) = u(Zj , t) exp(−iωmax t) s 00 h i |ωmax | = A0 sech A0 (j − j0 )a Q2
50
1
2.
(35)
P2 < 0 while the sign of Q2 depend on the value of η. For η < 0, then Q2 > 0 and the sign of the product P Q > 0, in this case (23) has the bright soliton solution
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Φj
2
0.2
For this sec-
io 12 n hp A(Zj , t) = A0 1 − σ 2 sech C2 σA0 (j − j0 )a ,
(42)
where C2 = −Q2 /2P2 = η/2a2 . The phase function ϕ(Zj , t) expression is p n p C2 A0 1 − σ 2 (j − j0 )a p io tanh C2 A0 σ(j − j0 )a
ϕ(Zj , t) =
h σ + tan−1 √ 1 − σ2
−
Q2 (3 − σ 2 )A20 t. (43) 2
As we have found in the first case, inserting Eqs. (42) and (43) into (31), then one can get easily the dark soliton solution in the form
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V.
MODULATIONAL INSTABILITY
(46)
To study the linear stability of the initial plane waves in the Bose-Hubbard chains, one can introduce a small perturbation δΦj (t) and seek a solution as Φj (t) = Φ0 + δΦj (t) exp[i(kaj − ωt)].
(47)
Inserting (47) into (8) and keeping in mind only the linear terms, one can get the following linear differential equation dδΦj + ωδΦj + cos(ka) δΦj+1 + δΦj−1 dt +i sin(ka) δΦj+1 − δΦj−1 + η 2|Φ0 |2 δΦj + Φ20 δΦ∗j = 0. (48) i
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where Ω4 = ωmax + η2 (3 − σ 2 )A20 . Eq. (44) is the dark soliton solution whose eigenfrequency Ω4 > ωmax is below the upper of the linear wave spectrum. In this second case, we also predict the existence of modulationnal instabilty and modulational stability of plane wave at the edge of the brillouin zone in presence of the interaction of two bosons at the same site depending of the sign of η.
ω = −2 cos(ka) − η|Φ0 |2 .
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Φj (t) = u(Zj , t) exp(−iωmax t) n hp io 12 = A0 1 − σ 2 sech C2 σA0 (j − j0 )a hp n p × exp i C2 A0 1 − σ 2 (j − j0 )a h p io σ tanh C2 A0 σ(j − j0 )a + tan−1 √ 1 − σ2 i −iΩ4 t , (44)
here a is the lattice constant, Φ0 is the initial amplitude, k and ω are respectively the wave number and the angular frequency of the plane wave. Inserting (45) into (8) one can have the following nonlinear dispersion relation
The general solution of (48) has the form
(45)
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Φj = Φ0 exp[i(kaj − ωt)],
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In this section, in order to study the modulational instability of (8), we adopt the standard linear stability analysis. First of all, we consider the propagation of wave signal along the Bose-Hubbard chains. Secondly, we find that (8) allows a constant amplitude solution which can take the form
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Ω − M11 M12 A 0 = , M21 Ω − M22 B 0
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(50)
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M11 = M22 = 2 sin(ka) sin Q M12 = 2 cos(ka) cos(Q) − 2 cos(ka) + η(|Φ0 |2 − Φ20 ) M21 = 2 cos(ka) cos(Q) − 2 cos(ka) + η(|Φ0 |2 + Φ20 ). (51) The condition for the existence of nontrivial solutions of (50) can be found by the following quadratic equation for the frequency Ω 2 Ω − M11 − M12 M21 = 0. (52)
δΦj (t) = A cos[Qj − Ωt] + iB sin[Qj − Ωt],
(49)
where k and Ω represent the wave number and the frequency of the perturbation, respectively. A and B stand for constant perturbations amplitudes. Inserting (49) into (48), we have the following linear equations for A and B
becomes negative then the perturbed wave can be unstable if the following condition is satisfied n o cos(ka) sin2 (Q/2) 2 cos(ka) sin2 (Q/2) − η|Φ0 |2 < 0.
(54)
(54) gives the modulational instability criterion of the Bose-Hubbard chains. From (54) one can get easily the threshold amplitude given by |Φ0,th |2 =
2 cos(ka) sin2 (Q/2) . η
(55)
Therefore, the modulational instability will take place if the initial amplitude |Φ0 |2 exceeds the threshold amplitude |Φ0,th |2 (|Φ0 |2 > |Φ0,th |2 ). By considering the fact that the right hand side of (52) becomes negative, two complex numbers are solutions. Along the same line, the corresponding growth rate of modulational (52) can be rewritten as follows n o instability is given as in refs [30–37, 40, 41, 43] by Ω21 = 8 cos(ka) sin2 (Q/2) 2 cos(ka) sin2 (Q/2) − η|Φ0 |2 . Γ(k, Q) = Q r Q (53) 2 sin . 2η|Φ0 |2 cos(ka) − 4 cos2 (ka) sin2 2 2 The modulational instability will be developed in the Bose-Hubbard chains if the right hand side of (53) (56)
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FIG. 4: (color online) The threshold amplitude for different negative values of the parameter η controlling the strength of the interaction in the lattice , where a = 1, Φ0 = 0.1: a) η = −0.01; b) η = −15; c) η = −35 and d) η = −80.
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To appreciate the effect of η, which is an interaction of two bosons at the same site in the lattice on the modulational instability criterion, let us plot the threshold amplitude for different values of η corresponding to attractive and repulsive interactions. For the case of repulsive interaction, the corresponding features are plotted in Fig. 4. From the threshold amplitude diagrams, the main remark observed here is that reducing the value of η leads to reduce values of the threshold amplitude. The consequence of this attitude, is an increasing of the instability region.
Remarkably, we realize in Fig. 5 (a) for wave number k[0; 0.5π] in the Brillouin zone that when the parameter η is non nil (η 6= 0), the plane wave is stable for any perturbation Q, this implies according to [42] that a dark-type localized mode may exist in the system. Therefore, from this observation we conclude that the stationary bright type intrinsic localized mode can not appear at the center of Brillouin zone. However the modulational instability begins to appear for k]0.5π; π] (see Fig. 5 (a)). Decreasing the value of η, we show in Fig. 5 (b) that, the shape of the region of modulational instability increases as an impact of the parameter η while its value is negative in the Bose-Hubbard chains. Furthermore,
In the corresponding stability/instability diagrams in the (k, Q) plane represented in Fig.5, the dark bluish area corresponds to a region where the nonlinear plane wave is stable under modulation whereas the region with bright yellowish orange area are regions where modulational instability is expected. The localized modes with dark bluish area physically can occur in this system when modulational instability does not appear. As we can see in Fig.5, the wave will be unstable in the regions of instability for k]0.5π; π] and Q]0; π].
we also notice that on varying the value of η from -35 to -80 and beyond this value, the domain of modulational instability changes dramatically and remains unchanged as we can see in Figs. 5 (c)-(d). For the case of attractive interaction, the threshold amplitude diagrams are represented in Fig.6. The fundamental behavior presented here, is that increasing the value of η leads to reduce values of the threshold amplitude and consequently an increasing of the instability regions. The corresponding stability/instability diagrams is portrayed in Fig.7, where the regions of instability have changed comparing to the previous case and becomes k[0; 0.5π[ and Q]0; π] for the unstable
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FIG. 5: (color online) Stability/instability diagrams in the (k, Q) plane for different values of the parameter η controlling the strength of the interaction in the lattice, where a = 1, Φ0 = 0.1: a) η = −0.5; b) η = −15; c) η = −35 and d) η = −80.
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wave.
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It is clearly illustrated in Fig. 6 (a) that, when the value of the interaction of two bosons is positive, the modulational instability can occur for k = 0 at the center of the Brillouin zone. It is important to mention here that the plane wave is stable for any perturbation Q at the edge of the Brillouin zone for k = π/a. When the value of η > 0 becomes sufficiently strong, the shape of the region of modulational instability changes significatively in Bose-Hubbard chains (see Fig. 6 (b) - (d)). For better understanding of the dependence of the modulation instability for k = 0 and k = π/a plane wave, we analyze the instability growth rates for different values of the parameter η. Two configurations shall be distinguished here. First of all, we consider the case for k = π/a and η < 0: In this case, we plot the instability growth rates for different values of η. Figure 8 portrays the growth rate of modulation wave for k = π/a. We find that, the instability growth rate may be affected by the parameter η controlling the strength of the interaction in the lattice. As we can see in Fig.8, the
instability region increases with a decreasing value of η, while the maximum value of the growth rate increases. For η = 0, we confirm here that the growth rate is nil. In this case there is not formation of bright envelope type localized mode as we described in Fig. 5. According to the modulational instability theory of carrier waves, the modulational instability is the best way to predict the existence of bright type localized modes. Our results confirm why the bright soliton solution can occur at the center of the Brillouin zone in the presence of the interaction η of two bosons at the same site when η < 0. Secondly, we consider the case for k = 0 and η > 0: We realize that when η > 0, the instability region increases while the maximum value of the growth rate increases as the value of η increases. In this case the maximum value of the growth rate remains constant as we observed in Fig.8, this case is not plotted here to avoid overloading the paper but it still identical to the case plotted in Fig.8 with the difference that the value of η is positive. Thus, the formation of bright type localized modes at the center of the Brillouin zone agreed with our analytical
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FIG. 6: (color online) The threshold amplitude for different positive values of the parameter η controlling the strength of the interaction in the lattice, where a = 1, Φ0 = 0.1: a) η = 0.5; b) η = 15; c) η = 35 and d) η = 80.
CONCLUSION
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VI.
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analysis which predicts the existence of the bright
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In this present work, quantum breathers and intrinsic localized excitation associated with modulational instability in 1D Bose-Hubbard boson chains have been studied.
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For the case of two bosons, by means of number state method in addition to the numerical diagonalization, we showed that the appearance of attractive and repulsive localized bands on the energy spectrum interpreted as quantum breathers can be controlled by varying the value of η. To study intrinsic localized modes in 1D BoseHubbard boson chains, we used the Ehrenfest theorem in the coherent state representation to obtain the equation of motion for the coherent-state amplitude. This equation is reduced to the NLS equation by the aid of multiple scales method combined with a quasi-discreteness approximation. We found that two types of intrinsic localized modes bright and dark can exist at the center and at the edges of Brillouin zone respectively in the system. The conditions
type localized modes.
of existence of these types of intrinsic localized modes are analyzed and our result reveals that they depend on the sign of the value of the interaction η. According to the modulational instability theory of carrier waves, the modulational instability is the best way to predict the existence of bright and dark type localized modes then, we have investigated the modulational instability analysis in the system to confirm the existence for bright and dark solitons and localized modes. As a result we found that modulational instability provided the appearance condition for bright solitons and localized modes. We also noticed that the shape of the region of modulational instability and the instability growth rates of the wave number k = 0 and of k = π/a can be more affected dramatically when the interaction of two bosons at the same site is included in the system. Our outcome certifies why the bright soliton can occur at the center and at the edge of the Brillouin zone which is in excellent agreement with the method of multiple scale in addition to a quasi-discreteness approximation analysis. The interaction of two bosons at the same site can
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η1 = 0.0 0.6
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η2 = − 15
0.4
0.2
0.2
0.4
Q/π
0.6
η4 = − 80
Acknowledgments 0.8
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0 0
be very significant in many physical systems, especially in quantum information. Such a theoretical work in quatum Bose-Hubbard boson chains needs more experimental investigation.
η3 = − 35
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The Growth Rate(‘Γ)
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FIG. 7: (color online) Stability/instability diagrams in the (k, Q) plane for different values of the parameter η controlling the strength of the interaction in the lattice, where a = 1, Φ0 = 0.1: a) η = 0.1; b) η = 15; c) η = 35 and d) η = 80.
[1] S. Flach and C. R. Willis, Discretes breathers Phys. Rep. 295 181-264 (1998) [2] S. Flach and A. V. Gorbach, Discrete breathers-advances in theory and applications Phys. Rep. 467 1 (2008) [3] R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators Nonlinearity 7 1623 (1994) [4] R. Sarkar and B. Dey, Energy localization
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FIG. 8: (color online) The growth rate versus the wave number of the perturbation Q for different values of η, where a = 1, Φ0 = 0.1, k = π/a and η < 0.
Zacharie Isidore Djoufack would like to thank the African Institute for Mathematical Sciences (AIMS) South Africa where a part of this work was done. The authors also acknowledge the referees for their worthy suggestions.
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Quantum breathers and intrinsic localized excitation associated with the Modulational instability in 1D Bose-Hubbard chain Z.I. Djoufack, E. Tala-Tebue, J-P Nguenang PII: DOI: Reference:
S1007-5704(18)30231-4 10.1016/j.cnsns.2018.07.018 CNSNS 4591
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
10 June 2017 15 June 2018 9 July 2018
Please cite this article as: Z.I. Djoufack, E. Tala-Tebue, J-P Nguenang, Quantum breathers and intrinsic localized excitation associated with the Modulational instability in 1D Bose-Hubbard chain, Communications in Nonlinear Science and Numerical Simulation (2018), doi: 10.1016/j.cnsns.2018.07.018
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• In this work, the energy spectrum and intrinsic localized modes associated with the modulational instability of boson chains can be described by a Bose-Hubbard model. By using of number states method combined with the numerical diagonalization, we show that the energy spectrum of the system exhibits bound states signature. With the help of multiple scales method in addition to a quasidiscreteness approximation, we obtain bright and dark type intrinsic localized modes at the center and at the edges of the Brillouin zone respectively and their appearance conditions. We found that the inclusion of interaction of two bosons at the same site can influence the appearance conditions of bright and dark solitons solution.Considering the fact that nowadays, the forthright way to predict the formation of intrinsic localized modes in nonlinear systems is the modulational instability. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in the Bose-Hubbard chain. It is found that the shape of the region of modulational instability and the instability growth rates of the plane wave can be more affected dramatically when the interaction of two bosons at the same site is involved in the system. Furthermore, we calculate the instability growth rates of plane wave at the center and at the edge of the Brillouin zone for different value of the interaction of two bosons to analyze their formation conditions which are in full agreement with the method of multiple scale in addition to a quasidiscreteness approximation analysis
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Highlights
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Quantum breathers and intrinsic localized excitation associated with the Modulational instability in 1D Bose-Hubbard chain Z. I. Djoufack
1,2 ∗
, E. Tala-Tebue 1 , and J-P Nguenang
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1 Fotso Victor University Institute of Technology, Department of Telecommunication and Network Engineering, P.O. Box 134 Bandjoun, University of Dschang, Cameroon 2 African Institute for Mathematical Sciences, 6 Melrose, Muizenberg, Cape Town 7945, South Africa and 3 Fundamental physics laboratory: Group of Nonlinear physics and Complex systems, Department of Physics, University of Douala, P.O. Box 24157, Douala, Cameroon (Dated: July 10, 2018)
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In this work, the energy spectrum and intrinsic localized modes associated with the modulational instability of boson chains is described by a Bose-Hubbard model. By using of number states method combined with the numerical diagonalization, we show that the energy spectrum of the system exhibits bound states signature. With the help of multiple scales method in addition to a quasidiscreteness approximation, we obtain bright and dark type intrinsic localized modes at the center and at the edges of the Brillouin zone respectively and their appearance conditions. We found that the inclusion of interaction of two bosons at the same site can influence the appearance conditions of bright and dark soliton solutions. Considering the fact that nowadays, the forthright way to predict the formation of intrinsic localized modes in nonlinear systems is the modulational instability. We investigate analytically, through the linear stability of plane wave solutions, the existence of localized structures in the Bose-Hubbard chain. It is found that the shape of the region of modulational instability and the instability growth rates of the plane wave can be more affected dramatically when the interaction of two bosons at the same site is involved in the system. Furthermore, we calculate the instability growth rates of plane wave at the center and at the edge of the Brillouin zone for different values of the interaction of two bosons to analyze their formation conditions which are in full agreement with the method of multiple scale in addition to a quasi-discreteness approximation analysis.
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Keywords: Quantum breathers; Bose-Hubbard model; Modulational instability and intrinsic localized modes.
INTRODUCTION
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PACS numbers: 05.45.Yv, 52.35.Sb, 42.65.Tg, 42.81.Dp
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Localized nonlinear excitation, generic time periodic and spatially localized solutions of the nonlinear systems with translational invariance due to the interplay between nonlinearity and discreteness are called discrete breathers [1, 2]. The existence of breather excitations in 1D lattices was first established by MacKay and Aubry [3]. Discrete breathers which play important roles in both energy storage and transport in various low dimension materials [4] have attracted much attention of intense theoretical and experimental subjects [5–10]. Experimentally, Discrete breathers have been observed in many physical systems such lattice vibrations in crystals, magnetic solids, Josephson junction, photonic crystals and antiferromagnetic structures [2], to mention a few. These latter studies demonstrate that intensive works have been investigated to master discrete breathers in classical nonlinear lattices. However, their quan-
∗ Corresponding
author: [email protected] (Zacharie Isidore Djoufack)
tum counterparts namely quantum breathers are less known nowadays and attract the attention of searchers in different domains such as mathematical and physical systems. In quantum domain, quantum breathers consist of superpositions of nearly degenerate many quanta bound states, characterized with long times to tunnel from one lattice site to another [11]. In the past, by means of numerical exact diagonalization, Wang et al. found the existence of quantum breathers in Klein Gordon atomic lattice model [12]. Furthermore, several works have been investigated on quantum properties of phonons [13–18]. Very recently, we found that two, four, and six-quanta quantum breathers exist in ferromagnetic spin chains using perturbation theory and exact numerical diagonalization [19, 20]. Using the similar technique namely, perturbation theory and numerical diagonalization, some works have shown the existence of quantum breathers in Bose-Hubbard model [21–23] and in 1D ultracold bosons in optical lattices [24]. A basic process in the theory of nonlinear wave is the modulational instability [25–29]. Modulational in-
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THE MODEL
The Hamiltonian of Bose-Hubbard discrete with bosonic operators is read: H1 = −
f h X j=1
i γa†j a†j aj aj + 1 (a†j aj+1 + aj a†j+1 ) , (1)
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where γ is the parameter controlling the strength of the interaction in the lattice and well known as an interaction of two bosons at the same site in the lattice, a†j and aj are the bosonic creation and annihilation operators. These operators satisfy the following commutation relations: [ai , a†j ] = δij , [a†i , a†j ] = [ai , aj ] = 0. While rescaling (1) by 1 , it can be written as f h i X f1 = H1 = − H ηa†j a†j aj aj + a†j aj+1 + aj a†j+1 ,(2) 1 j=1
where η = γ/1 . Due to the fact that the chain is tanslationally invariant, the Hamiltonian of this quantum system commutes with the number operator b = Pf a† aj , whose eigenvalue is n. N j=1 j
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stability is considered as an universal phenomenon that occurs in many domains such as optics, hydrodynamics, biology, to mention a few [30–35, 37, 38]. Modulational instability is defined as dynamical instabilities characterized by an exponential growth of amplitude perturbations which can occur during propagation under the interplay between nonlinearity and dispersion. Modulational instability in nonlinear lattice is the best way to predict the formation intrinsic localized modes [39, 40]. It is considered to some extent, a precursor to soliton formation. Kivshar and Peyrard were the first to establish that the modulational instability can be used to explore the generation of localized states in nonlinear lattices. They also showed that the discreteness can drastically modify the conditions for modulational instability [41]. Moreover, it was shown by Kivshar [42] that when the system displays modulational instability, only bright intrinsic localized mode exist. Daumont et al. predicted the evolution of a linear wave in the presence of noise, and also proved that modulational instability is the first step towards energy localization [43].
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However, in the case of the Bose-Hubbard model very few studies have been mostly devoted to analyze the non-equilibrium quantum dynamics of a Bose-Einstein condensate and density modulations associated with the dynamical instability [44, 45]. From the foregoing, it is clear that for better understanding of the generation of the bright soliton and the intrinsic localized mode in Bose-Hubbard chains, it is normal to improve investigation in the modulational instability. The Bose-Hubbard model is a very interesting nonlinear lattice which can help to better understand quantitatively the role played by the lattice interaction of two bosons at the same site and attracts our attention.
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The purpose of this paper is to explore quantum breathers and intrinsic localized excitation associated with the modulational instability in 1D Bose-Hubbard chains to analyze the influence of the interaction of two bosons at the same site on the properties of quantum breathers, intrinsic localized excitation and modulational instability in this nonlinear system. This paper is organized as follows: the BoseHubbard model is described in section II. In section III, we use the number state method combined with numerical diagonaliztion to prove that bound states can be controlled by the interaction between two bosons. In section IV, to predict the formation conditions of intrinsic localized modes, a linear analysis is carried out by means of the multiple scale method in addition to a quasi-discreteness approximation. Modulational instability/stability criterion and growth rates of plane waves are investigated in section V. Section VI concludes the paper.
III. INTERACTION BETWEEN TWO BOSONS IN THE BOSE-HUBBARD CHAIN
To study the two bosons’ case, we use a position state representation to describe the quantum states, | ψj =| n1 , n2 , · · · , nf , where P nj stands for the number of bosons at site j (n = j nj ). For instance, the state | 20100 · · · 0 represents a state with 2 bosons at the site 1, another boson at site 3 and nothing elsewhere. One can apply the translation operator to these states. This can be explained by the fact that chain is subjected to periodic boundary conditions. For a given number of bosons, each eigenstate is a linear combination of the number state with fixed n. For the sake of simplicity, we consider in this work an odd number of sites f . A general eigenfunction of these states for the Hamiltonian (2) is a Bloch wave. In the case for two bosons, this eigenfunction is read:
|ψ =
(f +1) 2
X j=1
f X Tb s−1 1 Cj | ψj = √ C1 | 20 · · · 0 τ f s=1
+C2
f X Tb s−1 s=1
+ · · · + C (f +1) 2
τ
f X Tb s−1 s=1
τ
| 110 · · · 0 (3)
| 10 · · · 01 ,
here Tb is the translation operator and k = 2πν/f , f −1 ik with ν ∈ {− f +1 represents the 2 , 2 } and τ = e eigenvalue of the translational operator Tb. To verify
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f +1 2
+τ
f +1 2
).
ENERGY SPECTRUM
0 -5
-10 -1
10
0
0.5
1
0
0.5
1
-0.5
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The eigenspectra of (4) plotted in Fig. 1 for different values of the parameter η is obtained by using the numerical diagonalization method. As we can see in Fig. 1(a), in this attractive case (η > 0), the isolated bound state is localized below the continuum band. From a physical picture applied to this quantum lattice, the isolated bound state represents the case where two bosons occupy the same site along the lattice and the probability of finding two bosons on the same site is high, confirming that the isolated bound state is a localized quantum sate (see Fig. 2). In the absence of the interaction of two bosons at the same site (η = 0), the energy spectrum in this case is only composed by the continuum band. Figure 1(b) shows the repulsive case for η < 0 with the black curve on top.
5
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where q1 = 1+τ 1 , q1∗ = 1+τ −1 and p = τ −
(a)
ENERGY SPECTRUM
that ψ | ψ = 1, it is important that C1 , C2 , ... must P (f +1) 2 | Cj |2 = 1. We can generbe normalized as j=1 ate an equivalence class of states with the aid of the translation operator Tb defined as Tba†j = a†j+1 Tb, so that Tb[n1 , n2 , · · · , nf ] = [nf , n1 , · · · , nf −1 ]. With the help of the basis given in (3), it is obvious to construct the following Hamiltonian matrix √ ∗ 2q1 0 0 0 0 0 0 √η 2q1 0 q1∗ 0 0 0 0 0 0 q1 0 0 0 0 0 0 .. .. .. 0 . . . 0 0 0 0 f1 = − H (4) . . . . . . . . . . 0 . . . . . 0 0 .. .. .. .. 0 . . . . q1∗ 0 0 0 0 0 0 0 q1 0 q1∗ 0 0 0 0 0 0 q1 p
5
0
-5 -1
-0.5
k/π
FIG. 1: (color online) Energy spectrum for two bosons’case for different values of η, f = 101: (a) η = 10 for attractive case; (b) η = −10 for repulsive case. 1
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LINEAR STABILITY ANALYSIS 0.8
aj |Φj i = Φj |Φj i,
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a†j |Φj i = Φ∗j |Φj i, ∞ 1 X (Φj )l √ |li. |Φj i = exp − |Φj |2 2 l! l=1
(5)
Φj in (5), represents the coherent amplitude of the operator aj for the system in the state |Φj i. The system state |Ψ(t)i in the coherent state representation, is defined by the product of the multimodes coherent state which can take the following form Y Φj , Ψ(t) = (6) j
(6) should be normalized as hΨ(t)|Ψ(t)i = 1. With the aid of Ehrenfest’s theorem [47], the dynamics of
Cj
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To describe the components of quantum state of the Bose-Hubbard chains, we adopt the Glauber’s coherent-state [46] defined as:
2
IV.
k/π (b)
0.6
0.4
0.2
0 0
20
40
60
Sites j
80
100
120
FIG. 2: square wave amplitudes of the eigenvectors in real space. Here η = 10, f = 101
bosons can be expressed in terms of the Heisenberg equation of motion for the Bose operator annihilation aj as i
dhaj i f1 ]i. = h[aj , H dt
(7)
ACCEPTED MANUSCRIPT 5 From (7), one can derive easily the equation of motion for the coherent state amplitude Φj read i
dΦj = −(Φj+1 + Φj−1 ) − η|Φj |2 Φj . dt
where A is an envelope function of variables τ and ζj which will be determined later. Inserting (16) into (15), one can get the following equation which stands for the boson linear dispersion relation ω(k) = −2 cos(ka).
(8)
Φj (t) = Φ
(1)
2
(τ, ζj , αj ) + Φ
(2)
(τ, ζj , αj ) ∞ X +··· = µ Φµj,j ,
(9)
µ=1
i
∂ (2) (2) (2) Φ + Φj,j−1 + Φj,j+1 ∂t j,j h i ∂ = i λ − Vg A(τ, ζj ) exp(iαj ), ∂ζj i
∂ (m) (m) (m) (m) Φ + Φj,j−1 + Φj,j+1 = γj,j , (m = 1, 2, 3 · · · ) ∂t j,j (10)
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with (1)
γj,j = 0,
λ=
dω = Vg = 2a sin(ka), dk
(19)
it is important to note here that λ is the group velocity. By solving (18) and considering the condition (19), (18) admits the special solution in the form (2)
Φj,j = B(τ, ζj ) exp(iαj ),
(20)
where B = B(τ, ζj ) is another function which will be determined in higher-order approximations. Next, if we set m = 3 in (10), considering the results previously obtained in (16), (17), (19) and (20) respectively, then we get ∂ (3) (3) (3) Φ + Φj,j−1 + Φj,j+1 ∂t j,j h ∂A i ∂2A =− i + a2 cos(ka) 2 + η|A|2 A exp(iαj ). ∂τ ∂ζj i
∂ (2) ∂ (1) a2 ∂ 2 (1) (1) Φj,j − i Φj,j − Φ + Φ j,j−1 j,j+1 ∂ζj ∂τ 2 ∂ζj2 (21) ∂ (1) (1) (1) (1) +a Φj,j−1 − Φj,j+1 − η|Φj,j |2 Φj,j , ∂ζj (13) In order to valid the theory, one must ensure that the secular term in the right hand side of (21) disap(4) pears. Therefore, one can obtain the following evoluγj,j = · · · . (14) tion form of nonlinear amplitude A(τ, ζj ) equation
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(3)
γj,j = iλ
∂ (1) ∂ (1) (1) Φj,j−1 − Φj,j+1 , (12) Φj,j + a ∂ζj ∂ζj
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(2)
γj,j = iλ
(11)
(18)
where Vg = dω/dk is the group velocity of the carrier waves. It should be noticed here that the term proportional to exp(iαj ) on the right hand side of Eq.(18) is a secular term which should be canceled in order to valid the theory [50]. Therefore, we get
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where is a small finite parameter whose represents the relative amplitude of the excitations, Φµj,j ≡ Φ(µ) (τ, ζj , αj ), ζj are slow variables defined by ζj = (ja − λt) and τ = 2 t respectively. αj = kja − ωt is the ”fast” variable, stands for the phase of the carrier wave number, a is lattice constant, k is the wave number and ω represents the frequency of the carrier wave. Inserting expansion (9) into (8), we obtain the following equations:
If we set m to 2 (m = 2) in (10), then we can get the second-order approximation equation given as
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Due to its nonlinearity and discreteness, Eq.(8) is not easy to solve exactly. This is why in this work, we adopt the method of multiple scales combined with a quasi-discreteness approximation [48, 49] to find solution of (8). In the treatment, we first set
(17)
m For the sake of simplicity, the expressions of γj,j (m = 4, 5, · · · ) are not written down completely here. For the lowest order of , which means the case where m = 1, one can obtain the following linear wave equation
i Taking
∂ (1) (1) (1) Φ + Φj,j−1 + Φj,j+1 = 0. ∂t j,j
into
account
the
fact
that
(15) (1)
Φj,j
=
(1) Φj,j (τ, ζj , αj ),
we get the general solution of (15) written in the following form (1)
Φj,j = A(τ, ζj ) exp(iαj ), αj = kja − ωt,
(16)
i
∂A ∂2A + P 2 + Q|A|2 A = 0, ∂τ ∂ζj
(22)
where P = a2 cos(ka) and Q = η. With the help of the following transformation A(τ, ζj ) = (1/ε)u(Zj , t), ζj = ε(ja − Vg ) = εZj and considering that τ = ε2 t, (22) be rewritten in the following form i
∂u ∂2u +P + Q|u|2 u = 0, ∂t ∂Zj2
(23)
where Zj = ja − Vg . Eq.(23) is the nonlinear Schr¨odinger equation (NLS). Depending on the sign
ACCEPTED MANUSCRIPT 6 whereas we employ the method of multiple scale combined with a quasi-discreteness approximation. In Heisenberg antiferromagnetic chains [52], analogous excitations have been found. Taking into account that there is a direct link between the formation of the bright type intrinsic localized modes and modulational instability of plane waves thus, we predict the appearance of modulationnal instability of plane wave at the center of the brillouin zone in presence of the interaction of two bosons at the same site when it takes positive value corresponding to an attractive case.
1. First case: k=0. In this case, we have
On the other hand, we have seen that in this first case that, due to its intrinsic nonlinearity, the Bose-Hubbard chains may support bright soliton solution. From physical picture, it is well known that the bright envelope soliton can propagate without losing energy. This implies that the bright soliton in Bose-Hubbard chains may be important in quantum system such as in quantum information.
d2 ω > 0, dk 2 ω = ωmin = −2, 1 00 P1 = P (k = 0) = ωmin = a2 , 2 Q1 = Q(k = 0) = η. Vg = 0,
(24) (25) (26)
For η < 0, we find that Q1 < 0 and the sign of the product P Q < 0 and (23) has the dark soliton solution given by
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(27)
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of product P Q, the solution of this equation can be localized bright soliton solution which has a vanishing amplitude at Zj 7−→ ∞ if the sign of product P Q > 0, i.e., P and Q have the same sign. The NLS equation has a dark soliton solution where the sign of the product P Q < 0. These configurations are discussed in this study to specify different types of intrinsic localized wave modes in nonlinear lattice model. The group velocity of the traveling wave is equal to zero at the center of the Brillouin zone for k = 0 and at the edges of the Brillouin zone for k = ±(π/a)
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P1 > 0 while the sign of Q1 depend on the value of η. We find that for η > 0, then Q1 > 0 and the sign of the product P Q is positive and (23) has the bright soliton solution s 00 h i ωmin A0 sech A0 (j − j0 )a (28) u= Q1 1 00 × exp[i ωmin A20 t + ϕ0 )], 2
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where A0 and ϕ0 are integral constants. j0 is an arbitrary integer. Thus we have
(29)
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Φj (t) = Φ1j,j (t) = u(Zj , t) exp(−iωmin t) s 00 h i ωmin = A0 sech A0 (j − j0 )a Q1 with
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× exp(−iΩ1 t + ϕ0 ),
Ω1 = ωmin − a2 A20 .
(30)
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Eq.(29) is the bright soliton solution which localized at the lattice site j = j0 and whose eigenfrequency Ω1 < ωmin is below the bottom of the linear wave spectrum. The site j = j0 depends on the initial exciting condition of the Bose-Hubbard model, is the central position of the localized mode. This means that the system is translational symmetry and we conclude that such intrinsic localized modes may be excited at any lattice site in the Bose-Hubbard model. The time evolution of the square of the coherent-state amplitude for this bright envelope soliton obtained at the center of the Brillouin zone is shown in Fig. 3 (a) The bright type intrinsic localized modes obtained in this first case is analogous to the one found in classical ferromagnetic chains with on-site anisotropy [51] where authors used the continuum approximation
u(Zj , t) = A(Zj , t) exp(iϕ(Zj , t)),
(31)
where A(Zj , t) is the amplitude function which is given as follows n hp io 12 A(Zj , t) = A0 1 − σ 2 sech C1 σA0 (j − j0 )a , (32)
with σ[0, 1], and stands for a parameter controlling the depth of the modulation of the amplitude function whereas C1 = −Q1 /2P1 = −η/2a2 . The expression of the phase function ϕ(Zj , t) is p n p ϕ(Zj , t) = C1 A0 1 − σ 2 (j − j0 )a h p io σ + tan−1 √ tanh C1 A0 σ(j − j0 )a 1 − σ2 Q1 − (3 − σ 2 )A20 t. (33) 2 Inserting Eqs.(32) and (33) into (31), then we have the dark soliton solution in the following form Φj (t) = u(Zj , t) exp(−iωmin t) n hp io 21 = A0 1 − σ 2 sech C1 σA0 (j − j0 )a hp n p × exp i C1 A0 1 − σ 2 (j − j0 )a h p io σ + tan−1 √ tanh C1 A0 σ(j − j0 )a 1 − σ2 i −iΩ2 t , (34)
where Ω2 = ωmin − η2 (3 − σ 2 )A20 . Eq.(34) stands for a dark type intrinsic localized mode in nonlinear lattice
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(a)
ond case, we get d2 ω < 0, dk 2 ω = ωmax = 2, 1 00 P2 = P (k = ±π/a) = ωmax = −a2 , 2 Q2 = Q(k = ±π/a) = η. Vg = 0,
0.1 0 2
50
1 Time
0 −50
0 Sites j
(b) u=
s
00 h i |ωmax | A0 sech A0 (j − j0 )a Q2 1 00 × exp[i ωmax A20 t + ϕ0 )], 2
(36) (37) (38)
(39)
thus it is not difficult to get
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Φj
2
0.2 0.1 0 2 0 −50
0 Sites j
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Time
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FIG. 3: Time evolution of the intensity of the coherentstate amplitude for the bright and dark soliton in the center of the Brillouin zone, where a = 1, f = 95 and σ = 0.5: a) k = 0 and η = 51.2; b) k = π/a and η = −39.8
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k = ±(π/a).
(40)
× exp(−iΩ3 t + ϕ0 ),
with
Ω3 = ωmax + a2 A20 .
(41)
Eq. (40) is the bright soliton solution whose eigenfrequency Ω3 > ωmax is below the upper of the linear wave spectrum. For η > 0, Q2 > 0 and the sign of the product P Q < 0 and (23) has the dark soliton solution given by (31) where the amplitude function is given as follows
models, whose eigenfrequency Ω2 lies the harmonic wave frequency band above. Then, we note here that this localized mode is a dark intrinsic localized nonlinear resonant mode in Bose-Hubbard system. The time evolution of the square of the coherent-state amplitude for this dark type intrinsic localized mode is shown in Fig. 3 (b) for a wave number k = 0 at the center of the Brillouin zone. As we can see in Fig. 3 (b), the time evolution of this nonlinear plane wave is symmetry and stable, thus such intrinsic localized modes can be excited at any lattice site in the Bose-Hubbard chains. We can predict here that the modulational instability will not take place in the system at the center of the Brillouin zone when the interaction of two bosons at the same site takes negative value corresponding to a repulsive case. Second case:
Φj (t) = u(Zj , t) exp(−iωmax t) s 00 h i |ωmax | = A0 sech A0 (j − j0 )a Q2
50
1
2.
(35)
P2 < 0 while the sign of Q2 depend on the value of η. For η < 0, then Q2 > 0 and the sign of the product P Q > 0, in this case (23) has the bright soliton solution
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Φj
2
0.2
For this sec-
io 12 n hp A(Zj , t) = A0 1 − σ 2 sech C2 σA0 (j − j0 )a ,
(42)
where C2 = −Q2 /2P2 = η/2a2 . The phase function ϕ(Zj , t) expression is p n p C2 A0 1 − σ 2 (j − j0 )a p io tanh C2 A0 σ(j − j0 )a
ϕ(Zj , t) =
h σ + tan−1 √ 1 − σ2
−
Q2 (3 − σ 2 )A20 t. (43) 2
As we have found in the first case, inserting Eqs. (42) and (43) into (31), then one can get easily the dark soliton solution in the form
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V.
MODULATIONAL INSTABILITY
(46)
To study the linear stability of the initial plane waves in the Bose-Hubbard chains, one can introduce a small perturbation δΦj (t) and seek a solution as Φj (t) = Φ0 + δΦj (t) exp[i(kaj − ωt)].
(47)
Inserting (47) into (8) and keeping in mind only the linear terms, one can get the following linear differential equation dδΦj + ωδΦj + cos(ka) δΦj+1 + δΦj−1 dt +i sin(ka) δΦj+1 − δΦj−1 + η 2|Φ0 |2 δΦj + Φ20 δΦ∗j = 0. (48) i
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where Ω4 = ωmax + η2 (3 − σ 2 )A20 . Eq. (44) is the dark soliton solution whose eigenfrequency Ω4 > ωmax is below the upper of the linear wave spectrum. In this second case, we also predict the existence of modulationnal instabilty and modulational stability of plane wave at the edge of the brillouin zone in presence of the interaction of two bosons at the same site depending of the sign of η.
ω = −2 cos(ka) − η|Φ0 |2 .
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Φj (t) = u(Zj , t) exp(−iωmax t) n hp io 12 = A0 1 − σ 2 sech C2 σA0 (j − j0 )a hp n p × exp i C2 A0 1 − σ 2 (j − j0 )a h p io σ tanh C2 A0 σ(j − j0 )a + tan−1 √ 1 − σ2 i −iΩ4 t , (44)
here a is the lattice constant, Φ0 is the initial amplitude, k and ω are respectively the wave number and the angular frequency of the plane wave. Inserting (45) into (8) one can have the following nonlinear dispersion relation
The general solution of (48) has the form
(45)
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Φj = Φ0 exp[i(kaj − ωt)],
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In this section, in order to study the modulational instability of (8), we adopt the standard linear stability analysis. First of all, we consider the propagation of wave signal along the Bose-Hubbard chains. Secondly, we find that (8) allows a constant amplitude solution which can take the form
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Ω − M11 M12 A 0 = , M21 Ω − M22 B 0
CE
with
(50)
AC
M11 = M22 = 2 sin(ka) sin Q M12 = 2 cos(ka) cos(Q) − 2 cos(ka) + η(|Φ0 |2 − Φ20 ) M21 = 2 cos(ka) cos(Q) − 2 cos(ka) + η(|Φ0 |2 + Φ20 ). (51) The condition for the existence of nontrivial solutions of (50) can be found by the following quadratic equation for the frequency Ω 2 Ω − M11 − M12 M21 = 0. (52)
δΦj (t) = A cos[Qj − Ωt] + iB sin[Qj − Ωt],
(49)
where k and Ω represent the wave number and the frequency of the perturbation, respectively. A and B stand for constant perturbations amplitudes. Inserting (49) into (48), we have the following linear equations for A and B
becomes negative then the perturbed wave can be unstable if the following condition is satisfied n o cos(ka) sin2 (Q/2) 2 cos(ka) sin2 (Q/2) − η|Φ0 |2 < 0.
(54)
(54) gives the modulational instability criterion of the Bose-Hubbard chains. From (54) one can get easily the threshold amplitude given by |Φ0,th |2 =
2 cos(ka) sin2 (Q/2) . η
(55)
Therefore, the modulational instability will take place if the initial amplitude |Φ0 |2 exceeds the threshold amplitude |Φ0,th |2 (|Φ0 |2 > |Φ0,th |2 ). By considering the fact that the right hand side of (52) becomes negative, two complex numbers are solutions. Along the same line, the corresponding growth rate of modulational (52) can be rewritten as follows n o instability is given as in refs [30–37, 40, 41, 43] by Ω21 = 8 cos(ka) sin2 (Q/2) 2 cos(ka) sin2 (Q/2) − η|Φ0 |2 . Γ(k, Q) = Q r Q (53) 2 sin . 2η|Φ0 |2 cos(ka) − 4 cos2 (ka) sin2 2 2 The modulational instability will be developed in the Bose-Hubbard chains if the right hand side of (53) (56)
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FIG. 4: (color online) The threshold amplitude for different negative values of the parameter η controlling the strength of the interaction in the lattice , where a = 1, Φ0 = 0.1: a) η = −0.01; b) η = −15; c) η = −35 and d) η = −80.
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CE
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ED
To appreciate the effect of η, which is an interaction of two bosons at the same site in the lattice on the modulational instability criterion, let us plot the threshold amplitude for different values of η corresponding to attractive and repulsive interactions. For the case of repulsive interaction, the corresponding features are plotted in Fig. 4. From the threshold amplitude diagrams, the main remark observed here is that reducing the value of η leads to reduce values of the threshold amplitude. The consequence of this attitude, is an increasing of the instability region.
Remarkably, we realize in Fig. 5 (a) for wave number k[0; 0.5π] in the Brillouin zone that when the parameter η is non nil (η 6= 0), the plane wave is stable for any perturbation Q, this implies according to [42] that a dark-type localized mode may exist in the system. Therefore, from this observation we conclude that the stationary bright type intrinsic localized mode can not appear at the center of Brillouin zone. However the modulational instability begins to appear for k]0.5π; π] (see Fig. 5 (a)). Decreasing the value of η, we show in Fig. 5 (b) that, the shape of the region of modulational instability increases as an impact of the parameter η while its value is negative in the Bose-Hubbard chains. Furthermore,
In the corresponding stability/instability diagrams in the (k, Q) plane represented in Fig.5, the dark bluish area corresponds to a region where the nonlinear plane wave is stable under modulation whereas the region with bright yellowish orange area are regions where modulational instability is expected. The localized modes with dark bluish area physically can occur in this system when modulational instability does not appear. As we can see in Fig.5, the wave will be unstable in the regions of instability for k]0.5π; π] and Q]0; π].
we also notice that on varying the value of η from -35 to -80 and beyond this value, the domain of modulational instability changes dramatically and remains unchanged as we can see in Figs. 5 (c)-(d). For the case of attractive interaction, the threshold amplitude diagrams are represented in Fig.6. The fundamental behavior presented here, is that increasing the value of η leads to reduce values of the threshold amplitude and consequently an increasing of the instability regions. The corresponding stability/instability diagrams is portrayed in Fig.7, where the regions of instability have changed comparing to the previous case and becomes k[0; 0.5π[ and Q]0; π] for the unstable
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FIG. 5: (color online) Stability/instability diagrams in the (k, Q) plane for different values of the parameter η controlling the strength of the interaction in the lattice, where a = 1, Φ0 = 0.1: a) η = −0.5; b) η = −15; c) η = −35 and d) η = −80.
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wave.
AC
CE
PT
It is clearly illustrated in Fig. 6 (a) that, when the value of the interaction of two bosons is positive, the modulational instability can occur for k = 0 at the center of the Brillouin zone. It is important to mention here that the plane wave is stable for any perturbation Q at the edge of the Brillouin zone for k = π/a. When the value of η > 0 becomes sufficiently strong, the shape of the region of modulational instability changes significatively in Bose-Hubbard chains (see Fig. 6 (b) - (d)). For better understanding of the dependence of the modulation instability for k = 0 and k = π/a plane wave, we analyze the instability growth rates for different values of the parameter η. Two configurations shall be distinguished here. First of all, we consider the case for k = π/a and η < 0: In this case, we plot the instability growth rates for different values of η. Figure 8 portrays the growth rate of modulation wave for k = π/a. We find that, the instability growth rate may be affected by the parameter η controlling the strength of the interaction in the lattice. As we can see in Fig.8, the
instability region increases with a decreasing value of η, while the maximum value of the growth rate increases. For η = 0, we confirm here that the growth rate is nil. In this case there is not formation of bright envelope type localized mode as we described in Fig. 5. According to the modulational instability theory of carrier waves, the modulational instability is the best way to predict the existence of bright type localized modes. Our results confirm why the bright soliton solution can occur at the center of the Brillouin zone in the presence of the interaction η of two bosons at the same site when η < 0. Secondly, we consider the case for k = 0 and η > 0: We realize that when η > 0, the instability region increases while the maximum value of the growth rate increases as the value of η increases. In this case the maximum value of the growth rate remains constant as we observed in Fig.8, this case is not plotted here to avoid overloading the paper but it still identical to the case plotted in Fig.8 with the difference that the value of η is positive. Thus, the formation of bright type localized modes at the center of the Brillouin zone agreed with our analytical
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FIG. 6: (color online) The threshold amplitude for different positive values of the parameter η controlling the strength of the interaction in the lattice, where a = 1, Φ0 = 0.1: a) η = 0.5; b) η = 15; c) η = 35 and d) η = 80.
CONCLUSION
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VI.
ED
analysis which predicts the existence of the bright
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In this present work, quantum breathers and intrinsic localized excitation associated with modulational instability in 1D Bose-Hubbard boson chains have been studied.
AC
For the case of two bosons, by means of number state method in addition to the numerical diagonalization, we showed that the appearance of attractive and repulsive localized bands on the energy spectrum interpreted as quantum breathers can be controlled by varying the value of η. To study intrinsic localized modes in 1D BoseHubbard boson chains, we used the Ehrenfest theorem in the coherent state representation to obtain the equation of motion for the coherent-state amplitude. This equation is reduced to the NLS equation by the aid of multiple scales method combined with a quasi-discreteness approximation. We found that two types of intrinsic localized modes bright and dark can exist at the center and at the edges of Brillouin zone respectively in the system. The conditions
type localized modes.
of existence of these types of intrinsic localized modes are analyzed and our result reveals that they depend on the sign of the value of the interaction η. According to the modulational instability theory of carrier waves, the modulational instability is the best way to predict the existence of bright and dark type localized modes then, we have investigated the modulational instability analysis in the system to confirm the existence for bright and dark solitons and localized modes. As a result we found that modulational instability provided the appearance condition for bright solitons and localized modes. We also noticed that the shape of the region of modulational instability and the instability growth rates of the wave number k = 0 and of k = π/a can be more affected dramatically when the interaction of two bosons at the same site is included in the system. Our outcome certifies why the bright soliton can occur at the center and at the edge of the Brillouin zone which is in excellent agreement with the method of multiple scale in addition to a quasi-discreteness approximation analysis. The interaction of two bosons at the same site can
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12
η1 = 0.0 0.6
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η2 = − 15
0.4
0.2
0.2
0.4
Q/π
0.6
η4 = − 80
Acknowledgments 0.8
1
CE
0 0
be very significant in many physical systems, especially in quantum information. Such a theoretical work in quatum Bose-Hubbard boson chains needs more experimental investigation.
η3 = − 35
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The Growth Rate(‘Γ)
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FIG. 7: (color online) Stability/instability diagrams in the (k, Q) plane for different values of the parameter η controlling the strength of the interaction in the lattice, where a = 1, Φ0 = 0.1: a) η = 0.1; b) η = 15; c) η = 35 and d) η = 80.
[1] S. Flach and C. R. Willis, Discretes breathers Phys. Rep. 295 181-264 (1998) [2] S. Flach and A. V. Gorbach, Discrete breathers-advances in theory and applications Phys. Rep. 467 1 (2008) [3] R. S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators Nonlinearity 7 1623 (1994) [4] R. Sarkar and B. Dey, Energy localization
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FIG. 8: (color online) The growth rate versus the wave number of the perturbation Q for different values of η, where a = 1, Φ0 = 0.1, k = π/a and η < 0.
Zacharie Isidore Djoufack would like to thank the African Institute for Mathematical Sciences (AIMS) South Africa where a part of this work was done. The authors also acknowledge the referees for their worthy suggestions.
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