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Matter wave compactons in deep optical lattices with strong nonlinearity management
M. Salerno1, F. Kh. Abdullaev2
1 Dipartimento di Fisica "E.R. Caianiello", CNISM and INFN - Gruppo Collegato di Salerno, Universita di Salerno, Via Giovanni Paolo II, 84084 Fisciano (SA), Italy
2Deparment of Physics, Kulliyyah of Science, International Islamic University Malaysia,
25200 Kuantan, Pahang, Malaysia
salerno@sa.infn.it
PACS 42.65.-k, 42.81.Dp, 03.75.Lm DOI 10.17586/2220-8054-2015-6-6-742-750
Compacton matter waves are considered in Bose-Einstein condensates (BEC) and in binary BEC mixtures, trapped in deep optical lattices in the presence of strong and rapid periodic time modulations of the atomic scattering length, are considered. For this, we derive vector averaged discrete nonlinear Schrodinger equations (DNLSE) and show that compacton solutions of different types can exist as stable excitations. Stability properties are studied by linear analysis and by direct numerical integrations of the DNLSE system.
Keywords: Bose-Einstein condensates, wave compactons, solitons, nonlinear Schrodinger equation.
Received: 1 November 2015
1. Introduction
Periodic management of nonlinear wave system parameters is a very attractive technique for the generation of solitons with new types of properties [1]. Examples of the management technique in continuous systems are the dispersion management of solitons in optical fibers which allows one to improve communication capacities [2], and the nonlinearity management of 2D and 3D Bose-Einstein condensates (BEC) or optically layered media which provides partial stabilization against collapse in the case of attractive interatomic interactions [3]. In discrete systems, the diffraction management technique was used to generate spatially discrete solitons with novel properties [4, 5] which have recently been observed in experiments [5]. The suppression of the inter-well tunneling was experimentally observed for light propagation in waveguide arrays [6] and for BEC's in strongly driven optical lattices [7]. In these cases, however, the system is typically subjected either to resonant modulations of the dispersion (coupling between waveguides for the case of light propagation) or to external linear forces (optical lattice shaking in the BEC case). The inhibition of the inter-well tunneling, however, may also become possible in the presence of fast periodic time variations in the nonlinearity, the so called strong nonlinearity management (SNLM). In this case, new phenomena can arise, such as strong localization and formation of (discrete counterparts to) the so called compactons [8] e.g. localized nonlinear waves with compact support.
The aim of the present work is investigate discrete compactons solutions of the discrete nonlinear Schrodinger equation (DNLS) system [9] subjected to SNLM both for single component BEC and for vectorial (binary) BEC mixtures. Contrary to ordinary solitons, the amplitude of a compacton was shown to reduce exactly to zero outside of the localizing domain, implying the total suppression of the inter-well tunneling at the compacton edges. For this, we derive the averaged DNLSE and show that compacton solutions of different
types can exist as stable excitations [10]. The stability properties are studied by linear analysis and by direct numerical integrations of the DNLSE system. Similar results will also be derived for arrays of binary BEC mixtures described by the vector DNLSE equation in the presence of inter-species scattering length SNLM.
2. The model
The dynamical lattice considered here is the well known DNLSE [9]:
iun + K(un+1 + Un-1) + (yo + Y (i))|«n|2«n = 0, (1)
which serves as a prototypical model for both matter waves in BEC arrays and for light propagation in arrays of optical waveguides. In the BEC context, k quantifies the coupling (tunneling of matter) between adjacent wells of the optical lattice, t represents the time and Y0 and y(t) represent the constant and the modulated part of the interatomic interaction (nonlinearity), respectively. In this case the management corresponds to the periodically variable in time the atomic scattering length which can be achieved by the Feshbach resonance technique. In the optical context, the time t should be replaced by the propagation distance z, k quantifies the coupling between adjacent waveguides, and the nonlinear management consists of a periodic spatially variable Kerr nonlinearity around a constant value Y0. In the following, we shall refer to the BEC context and assume y(t) to be a periodic, Y(t) = y(t + T), and rapidly varying function of time of the form: y(t) = -¡r cos(Q|), with e ^ 1 and T = 2n/n the period.
To investigate the existence of discrete compacton solitons in this model, we shall derive averaged equations over rapid modulations, using the method developed in [11]. Following this approach, we introduce the new variables vn related to the field un as:
1 rt t
Un(t) = vn(t)eir(t)|vn|2(t), r(t) = - / dtYi(-). (2)
e Jo e
Substituting this expression into Eq.(1) and averaging the resulting equation over the period of the rapid modulation, we obtain:
ivn,t = -aKVn[(vn+ivn + vn+ivn)Ji(a0+) + (vn-ivn + vn_ivn)Ji(a6L)] - (3)
K[Vn+lJoM+) + vn-iJoM-)] - Yo|vn|2vn,
where Jn(x) is the Bessel function of order n, 9± = |vn±i|2 — |vn|2 and a = Yi/K. This modified DNLSE has essentially nonlinear neighbor-neighbor interactions and can be put in Hamiltonian form ivvn = JH-, with the following Hamiltonian:
Hav = — ^(KJo(a0+) [vn+ivn + vn+ivn] + Y20 |vn|4}. n
For small a0+ the function J0 can be expanded in a series giving the same averaged Hamiltonian for the DNLSE obtained in [12] in the limit of weak nonlinearity management.
3. Compactly-supported localized modes
In this section, we demonstrate the existence of exact compactons in this averaged system. We remark that compacton solutions were initially reported as a "mathematical curiosity" of somewhat artificial variants of the DNLSE where linear dispersion is absent [13], by analogy to their continuum siblings [8]. The present setting, however, is in some sense unique in that linear dispersion is not, generally speaking, absent. In fact, there is a linear spectrum for the background state in the linearization of even these compact solutions and
Fig. 1. Typical examples for k = 0.5, a = 1 of compact localized mode solutions of Eq. (4) (top panels) and of the spectral plane (Àr, Àj) of their linearization eigenvalues À = Àr + ¿Àj. 1st column: on-site, 2nd column: intersite in-phase compacton. Remarkably, all solutions are spectrally stable.
can be analytically shown to extend from [—^ — 2k, — ^ + 2k] and from — 2k, ^ + 2k]. Yet, there exist particular amplitude values which, in essence, completely inhibit the inter-well tunneling to the nearest neighbors and hence, enable the formation of such compact structures.
In this regard, we seek stationary solutions of the form vn = Ane-vi, for which Eq. (4) becomes:
Mn + YoA + K(An+i JoM+) + An-i Jo(a0-)) + 2aKAn[An+i AnJi(ad+) + An-iAnJi(ad-)] = 0, (4)
and for conditions of tunneling suppression at the last site of vanishing amplitude (edge of the compacton) denoted as n0 below. In the setting of Eq. (4), this directly establishes that:
Jo(a|uno+i|2) = 0 ^ |uno+i|2 = 2.4048/a, (5)
which yields the solution based on the first zero of the Bessel function for the "boundary" of the compactly supported site. Then, for ^ = — y0 |«no+i|2, both the condition for compact support at n0 ± 1, and the equation for n = n0 are satisfied. Hence, Eq. (5) yields a singlesite discrete compacton, which linearization illustrates to be stable (both the solution and its typical linearization are shown in Fig.1).
One can then generalize this type of consideration to two-sites, which are either in phase (2nd column of Fig.1) or out-of-phase (the so-called twisted modes of the 1st column of Fig.2). The only thing that changes here is that in order to satisfy the equation at the non-vanishing sites,
^ = -k - T0K0+1I2 ^ = k - Yo|«no+l|2,
(6)
Fig. 2. Compacton solutions of Eq.(4)1st column: inter-site out-of-phase mode , 2nd column: symmetric 3-site compacton.
respectively for the in-phase and out-of-phase two-site modes. We stress that these are exact solutions for the reduced system. Surprisingly, and completely contrary to what is the case for the DNLSE, both of these solutions are spectrally stable, as shown in Fig.2.
Moreover, with some additional effort, one can generalize these considerations to an arbitrary number of sites. As a typical example, a 3 site mode with amplitudes (..., 0, Ai, A2, Ai, 0,...) will satisfy, in addition to the "no tunneling condition" J0(aAi) = 0, the constraints:
juAi + ka? Jo(a(A2 - A?)) + 2anA\A2Ji(a(A? - A\)) + loA\ = 0, ^A? + kai Jo(a(A? - A?)) + 4aKA\AiJi(a(A\ - A?)) + y0a3 = 0
which can be easily solved to yield a solution as the one shown in the 2nd column of Fig.2. Even such more complex solutions which would be highly unstable in the DNLSE are dynamically robust in the present setting. To examine the full nonlinear dynamical stability of these solutions, we considered them as initial conditions both in the averaged Eq. (4), as well as in the full Eq. (1). The results are shown in Fig.3. The single-site results are shown (similar findings were obtained for other modes). The top panel (of large color-bar amplitude) shows the space-time contour map of the solution modulus, while the bottom panel (of small colorbar amplitude) illustrates the deviation from the original solution. To further ensure robustness, a uniformly-distributed random perturbation of small amplitude was added to the original solution. It can clearly be seen that in all cases, both in the averaged equation and in the original system of Eq. (1), the relevant perturbation stays uniformly bounded and never exceeds 2% of the solution amplitude. The waveforms remain remarkably localized in their compact shape (after a transient stage of shedding off small
Fig. 3. Time evolution of single-site compacton solution. The top panel shows the evolution obtained from the averaged equation in Eq. (4), while the bottom panel refers to the numerical integration of the original DNLSE system in Eq. (1).
Fig. 4. Time evolution of a perturbed 3-site compacton for к =1 decaying into a single-site compacton for e = 0.1 (top panel) and remaining stable for e = 0.025 (bottom panel).
amplitude wavepackets). Notice that for Eq. (1), 7(t) = 1 + ^ cos(t/e), with e = 0.1 was used.
It should, however, be noted that if one departs from the regime of averaging validity of the averaging, an interesting deviation from the above behavior (and stability) may ensue. An example of this is shown in the panels of Fig. 4. In this case, the 3-site solution was initialized in Eq. (1) with e = 0.1 in the top panel, while e = 0.025 in the bottom one. While in the latter case, the above-argued robustness of the averaged modes was observed, in the former one, the apparent lack thereof was clearly due to the use of an e outside of the regime of applicability for averaging approximation. Nevertheless, the resulting evolution confirms the general preference of the system towards settling in compact modes, since despite the large coupling k =1 used in this case, the evolution asymptotes to an essentially single-site solution.
4. Discrete compactons of binary BEC mixtures under SNLM
The above results can be extended to the case of two-component atomic BEC in a deep OL. In this case, the tight binding approximation leads to the following vector DNLSE [14]:
Hn = -Ki(«n+i + Un-i) - (Yi|Un|2 + Yi2K|?)Wn,
(7)
iVn = K2(vn+i + Vn-i) - (Yl2 |Un|2 + Y2|vn|2)vn,
where the overdot stands for time derivative, the coefficients Kj, i = 1, 2, are related to the tunneling rates of atoms between neighboring wells of the optical lattice and y12, i =1, 2 are nonlinear coefficients related to the inter-species (ai2) and intra-species (a^, i = 1, 2) scattering lengths, respectively. Eq. (7) also arises in nonlinear optics where it models the propagation of an electric field in an array of optical waveguides with variable Kerr nonlinearity. In this context, the roles of the nonlinear coefficients 7^ correspond to self- and cross-phase modulations of the electric field components, respectively [15,16]. Notice that the above two component DNLSE has the Hamiltonian form x, = ^H/^x, with
Xn Un,vn
and the Hamiltonian H given by:
H = - ^ [(Ki«n+1«n + K2Vn+1 < + c.c.) + 1 (7i|«n|4 + 72KI4) + 7i2|«n|2K|2 . (8)
Here, the * stands for the complex conjugation and c.c. denotes the complex conjugate of the expression in the parenthesis. Also, notice that the number of atoms N = |xn|2, Xn = un,vn is conserved for each component. Below, we concentrate on the case of BEC mixtures with fixed (not necessarily equal) intra-species nonlinearities and subjected to strong nonlinear inter-species management of the form:
Yi2(t) = YÏ? + YÍ2)(í) = yí2) + ^ cos(Q*), (9)
e e
with Yi2), Yi2) real constants and e a small parameter controlling the strength of the modulation as before (strong nonlinearity management corresponding to e ^ 1 with Q, y ( 2) ~ O(l)). We remark that although we have considered for simplicity in Eq. (9) a sinusoidal modulation of period T = 2n/Q in the fast time variable t = t/e, results can be extended easily to other types of periodic Y(2) (t) functions. This management setting appears to be quite convenient in experiments since it involves changing only one parameter (for the case of intra-species SNLM see [17]). Moreover, the averaged equations obtained in this case are physically more interesting because nonlinear dispersion terms that involve both components
could lead, in principle, to novel types of binary compactons. Analogously to what was done for the single component case above, we use the averaging method to eliminate the fast time, t = t/e, dependence. In this respect, we perform the following transformation:
un = Uneir|V"12, vn = Vneir|U"12, (10)
where r denotes the antiderivatives of Yi2)(t), e.g. ri2(T) = ^(r- JJ cos^t'^t'. By substituting Eq.(10) into Eqs.(7) we obtain:
iUUn = iK2r(T)Un[Vn*Xi — V;X*] — KiX2 — (Yi |Un|2 + Y02|Vn|2)Un, (11)
iV; = ¿kiT(t )Vn[UnX2 — UnX2 ] — K2Xi — (Y2|Vn|2 + Y(02)|Un|2)Vn, (12)
where Xi = Un+ieire+ + Un-ie*™-, X2 = Vn+ieire+ + Vn-ie^- and with 0± denoting the quantities:
e± = |Un±i|2 — |Un|2, e± = |Vn±i|2 — |K|2. (13)
The average over the rapid modulation in these equations can easily be done with the help of the relations:
< e±ir(T)0± > = J0M±)
^^ (14)
< r(T)e±ir(T)0± > = ±iaJi(a0±),
where the angular bracket (■) denotes the average with respect to the fast time, e.g. < F >= (1/T) J; FdT, while J0, Ji are Bessel functions [18] of the first kind for the zero-th and first order respectively, and the parameter a is given by:
a = Y&M (15)
The system of averaged equations is then obtained as:
iUn = —aK2Un
Ji(a0+) (vx+i + VnVnVi) + Ji(a^-) (Vn*Vn-i + VnVn-
— Ki [Un+iJ0M+) + Un-iJ0M-)] — Yi|Un|2 + Y^lVnl
Un
iV^ = —aKiVn
Ji(a0+) (unUn+i + UnUn+i) + Ji(a0-) (U;Un-i + UnU;-)
— K2 [Vn+iJ0(a0+) + Vn-iJ0(a0-)] — Y2|Vn|2 + Yu^2
V„,
(16)
(17)
Here, we only report on some interesting consequences implied by these equations for the simplest case of a bright-bright compacton, which can be searched in the form of for stationary states:
Un = Ane-^Vn = Bn(18) with chemical potentials of the two atomic species. Substituting these expressions
into Eq. (16,17), one gets the following stationary equations:
+ (YiAn + Y(2)Bn An) + Ki[An+i J0(a0+) +
An-i J0(a02-)] + 2aK2A;Bn[Bn+i Ji(a0+) + (19)
Bn-iJiM-)] = 0,
2
Bn + (Y2Bn + Y(02)AnBn) + K2[Bn+i J0M+) +
Bn-i J0(a0-)] + 2aKiAnBn[An+i Ji(a0+) +
An-iJi(a02-)] = 0,
to be solved for the chemical potentials and amplitudes An, Bn of the compacton modes. The compact nature of the solution (A¿, B = 0 outside a finite (small) range of sites), allows one to truncate the above infinite system into a finite number of relations between the above variables, which can be solved exactly.
4.1. Bright-Bright compactons
To search for bright-bright (B-B) compacton solutions we need to look for the last sites of vanishing amplitude, i.e. n0 ± 1, where the vanishing of the tunneling rate is realized. For a single site B-B compacton, we assume An0 = a, Bn0 = b, An0±j = 0, Bn0±j = 0 for all j ^ 1. Substituting this ansatz in Eqs.(19,20), we obtain the corresponding condition for the compacton existence as:
J0(aa2) = 0, a2 = /a,
Jo(a2b2) = 0, b2 = 6/a (21)
where £2 are zeros (not necessary equal) of the Bessel function J0. This condition, together with:
= -Yia2 - Yi2b2, ^ = -Y2b2 - Yi2a2, (22)
gives us the single site B-B compacton pair.
Notice that these equations are the same as for intra-species management case. A big difference with the intra-species management case, however, is the fact that now, the numbers of atoms in the two components of a single site compacton cannot be arbitrary but must be related as follows:
t = t • (23)
This relation implies that for a given number of particles in the first component N = a2 (controlled by the parameter a), only a discrete set of values for N2 are permitted. It is interesting to see if such a state can emerge from generic initial conditions (for example Gaussians). Further research is currently underway to see if this approach is feasible.
5. Conclusions
In conclusion, we have shown that in the SNLM limit stable discrete compactons can exist both in one-dimensional BEC trapped in deep optical lattices and in binary BEC mixtures. In this last case, the variety of the compact excitations is larger and includes also dark components in the form of stable bright-dark or dark-dark compactons. The lack of interacting tails in these excitations could be of interest for practical applications. In particular, in the field of nonlinear optics, this could permit one to achieve a maximum rate of information transfer signal in waveguide arrays without disturbing interferences. The solutions considered herein could also lead to experimental observation of discrete compactons both in BEC arrays and in arrays of binary BEC mixtures subjected to SNLM.
Acknowledgements
MS acknowledge partial support from the Ministero delllstruzione, dellUniversit e della Ricerca (MIUR) through a PRIN (Programmi di Ricerca Scientifica di Rilevante Interesse Nazionale) 2010 - 2011 initiative (grant # 2010HXAW77-005).
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