Alternative routes to the 1D discrete nonpolynomial Schrödinger equation

Boris MalomedDept. of Physical Electronics, Faculty of Engineering, Tel Aviv

University, Tel Aviv, IsraelGoran Gligorić and Ljupco Hadžievski

Vinća Institute of Nuclear Sciences, Belgrade, SerbiaAleksandra Maluckov

Faculty of Sciences and Mathematics, University of Niš, Niš, SerbiaLuca Salasnich

Department of Physics “Galileo Galilei”Universitá di Padova, Padua, Italy

The topic of the work: approximation of the Gross-Pitaevskii equation for BEC by means of discrete models, in the presence of a deep optical lattice. We report derivation of a new model, and its comparison with the previously know one. A simulating discussion with Dmitry

Pelinovsky is appreciated .Some revious works on the topic of the discrete approximation for BEC:

A. Trombettoni and A. Smerzi, Phys. Rev. Lett. 86, 2353 (2001); F.Kh. Abdullaev, B. B. Baizakov, S.A. Darmanyan, V.V. Konotop & M. Salerno, Phys. Rev. A 64, 043606 (2001) ;

G.L. Alfimov, P. G. Kevrekidis, V.V. Konotop & M. Salerno, Phys. Rev. E 66, 046608 (2002) ;

R. Carretero-González & K. Promislow, Phys. Rev. A 66, 033610 (2002) ;N.K. Efremidis & D.N. Christodoulides, ibid. 67, 063608 (2003);M. A. Porter, R. Carretero-González, P. G. Kevrekidis & B. A. Malomed, Chaos 15, 015115 (2005);G. Gligorić, A. Maluckov, Lj. Hadžievski & B.A. Malomed, ibid. 78, 063615 (2008).

The starting point: the rescaled three-dimensional (3D) Gross-Pitaevskii equation (GPE) for the normalized mean-field wave function. The GPE includes the transverse parabolic trapping potential and the longitudinal

(axial) optical-lattice (OL) periodic potential:

2 2 2 20

1 1cos(2 ) 2 | | ,2 2

where 2 / , / , is the trapping

frequency, is the number of atoms, and is the . The repulisve/attractive interactions

s

s

i V qz x yt

Na a a m

N ascattering length

between atoms correspond to 0/ 0, respectively.

The first (“traditional”) way of the derivation of the 1D discrete model: start by reducing the dimension of the continual equation from 3 to 1, and then discretize the effective 1D equation.

2 2

2

2

The reduction of the dimension from 3 to 1 is carriedout through the following :

1, , , exp ( , ),2 ( , ),

with normalization | ( , ) | 1.

The resulting

ansatz

x yx y z t f z tz tz t

f z t dz

nonpolynomial S

2 2

02 2

(NPSE)[L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 65, 043614 (2002)]:

1 1 (3 / 2) | |cos 2 .2 1 | |

This dimensional equation admits the descri

chroedinger equation

f fi V qz ft z f

one

2cr

ption of the

if < 0, at the : (| | ) 1/ .collapse critical density f

The discretization of the one-dimensional NPSE leads to the DNPSE [A. Maluckov, Lj. Hadžievski, B. A. Malomed, and L. Salasnich, Phys. Rev. A 78, 013616 (2008)], through the expansion over a set of localized wave functions

21 12

2

2

(e.g., Wannier functions):

( , ) ( ) ( ).

The result:

1 1 1 | |2 1 | |

| |.

1 | |

This also admits the possibility of the

n nn

nn n n n

n

nn

n

f z t f t W z

dfi g f f C f fdt g f

g ff

g f

discrete equation

2

for 0, at the of the on-site

density, | | 1/ .n

collapse g critical value

f g

The new (“second”) route to the derivation of the 1D discrete model: first perform the discretization of the underlying 3D GPE in the direction of z, and then perform the reduction of the dimension. The two operations are expected to be non-communitative.

2 2 2

21 1

Assume the expansion of the 3 wave function:

( , , , ) ( , , ) ( ).

This leads to the - equation,1 12 2

2 | | .

n nn

nn

n n n n

D

x y z t x y t W z

semi discrete

i x ytC g

The reduction of the semi-discrete equation to the 1D discrete form.

2 2

2

The dimension-reducing :

1( , , ) exp ( , ).2 ( , )( , )

The substitution of the ansatz into the semi-discreteequations and averaging in the plane of , leadsthe following sys

n nnn

ansatz

x yx y t f z tz tz t

x y

tem of coupled equations for thecomplex 1D discrete time-dependent wave function,

( ), and the real transverse width, (t), that may also be time-dependent functions.n nf t

At the first glance, this system seems very complex, and completely different from the DNPSE equation for the discrete wave function, fn(t), which was derived by means of the “traditional approach”. Our objective is to find and compare families of fundamental discrete solitons in both models, the “traditional simple”, and “new complex” ones.

The distinction between the “traditional” and “new” models is also seen in the difference between their Hamiltonians, while they share the same expression for the norm (P):

2

2 2 2old 1

* 2 1new 2 2 2

1

* * 41 1 2

| | ,

| | 1 | | | | ,

1 1 22

| | .2

{

}

nn

n n n nn

n nn n n

n n n n

n n n n nn

P f

H C f f g f f

H f f C

gf f f f f

First, we consider the modulational instability of CW (uniform) waves, fn(t) = U exp(-iμt), in the case of the attractive nonlinearity, g < 0. The CW solutions are unstable in regions below the respective curves (q is the wavenumber of the modulational perturbations):

Comparison of the shapes of unstaggered onsite-centered discrete solitons: transverse widths (σn) at the central site of the soliton, and two sites adjacent to the center, in the two models, for C = 0.2 and C = 0.8. Chains of symbols: the “old” model; curves: the “new” one.

The same comparison for unstaggered intersite-centered solitons:

Stationary soliton solutions with chemical potential μ are looked for as fn = Fn exp(-iμt), with real Fn, for two soliton families: onsite-centered and intersite-centered ones. Then, the families may be described by dependences of the

norm vs. the chemical potential ,

2( ) ( ).nn

P F

The comparison of the P(μ) characteristics for the family of onsite-centered solitons, at C = 0.2 and C = 0.8. Note that the VK stability criterion, dP/dμ < 0, generally, does not apply to the model with the nonpolynomial nonlinearity.

The same comparison for families of intersite-centered solitons, also for C = 0.2 and C = 0.8:

A similar comparison of values of the free energy for both models, G = H – μP, again for C = 0.2 and C = 0.8. Left and right curves in both panels pertain to on-site and inter-site solitons, respectively:

The comparison of dynamical properties of the discrete solitons in both models. In particular, real eigenvalues (“ev”), accounting for the instability of a part of the family of the onsite-centered solitons (a) and of the intersite-centered soliton family (b), at C = 0.8, as functions of μ, have the following form in both models (the intersite solitons are completely unstable in both models; onsite solitons have their stability region):

CONCLUSIONSThere are two alternative ways to derive the 1D discrete equation for BEC trapped in the combination of the tight transverse confining

potential and deep axial optical lattice :)1( First, reduce the dimension from 3 to 1,

arriving at the nonpolynomial NLS equation, and then subject it to the discretization; or

)2( First, apply the discretization to the 3D equation, reducing it to a semi-discrete form, and then eliminate (by averaging) the two transverse dimensions.

Because the reduction of the dimension and the discretization do not commute, these alternative routes of the derivation lead to two models which seem completely different: the earlier known discrete nonpolynomial NLS equation (obtained by means of the former method), or the model obtained by means of the latter method, which, apparently, has a much more complex form.

However, despite the very different form of the two models, they produce nearly identical results for the shape of unstaggered discrete solitons of both the onsite- and intersite-centered types, and virtually identical conclusions about their stability.

Additional numerical analysis demonstrates that the character of the collapse in both discrete models is also essentially the same (note that both models are capable to predict the collapse in the framework of the 1D discrete approximation), as well as regions of the mobility of the solitons.

The general conclusion :The approximation of the dynamical behavior of the 3D condensate in the present setting by the 1D discrete model is reliable, as two very different models yield almost identical eventual results.