Stability and collapse of localized solutions of the controlled three-dimensional Gross-Pitaevskii equation
On the basis of recent investigations, a newly developed analytical procedure is used for constructing a wide class of localized solutions of the controlled three-dimensional (3D) Gross-Pitaevskii equation (GPE) that governs the dynamics of Bose-Einstein condensates (BECs). The controlled 3D GPE is decomposed into a two-dimensional (2D) linear Schrödinger equation and a one-dimensional (1D) nonlinear Schrödinger equation, constrained by a variational condition for the controlling potential. Then, the above class of localized solutions are constructed as the product of the solutions of the transverse and longitudinal equations. On the basis of these exact 3D analytical solutions, a stability analysis is carried out, focusing our attention on the physical conditions for having collapsing or non-collapsing solutions.
pacs:02.30.Yy, 03.75.Lm, 67.85.Hj
About 85 years ago, the seminal work of Bose Bose () opened up the study of the statistical properties of bosons in ultra-cold quantum systems. Bose’s idea was further developed by Einstein Einstein (), leading to the theoretical prediction of the condensation of atoms in the lowest quantum state below a certain temperature. The idea of Bose-Einstein of atom condensation in the ground state has been experimentally verified in a dilute gas composed of atoms Ketterle (); Anderson (); Bradley (); Davis (). The dynamics of nonlinearly interacting bosons in ultra-low temperature gases is governed by the Gross-Pitaevskii equation (GPE), which is an extension of the nonlinear Schrödinger equation (NLSE) by including the confining potential and inter-atomic interactions Gross-Pitaevskii (). The GPE, without the external potential, admits localized solutions in the form of one-dimensional dark and bright solitons, as well as radially symmetric vortex structures. Nonlinear localized excitations involving BECs arise due to the balance between the spatial dispersion of matter waves and nonlinearities caused by repulsive or attractive inter-atomic interactions in BECs. Recent observations r1a (); r1b (); r1c (); r1d (); r2a (); r2b (); r2c (); r2d () conclusively demonstrated the existence of bright r2a (); r2b (); r2c (); r2d () and dark/grey r1a (); r1b (); r1c (); r1d () matter wave solitons and quantum vortices quantum-vortices ().
Although the area of investigations of localized solutions of the GPE is quite fascinating, most of the theoretical results deals with approximate solutions in 3D or in reduced geometries 5 (). They are well supported by suitable numerical evaluations 6 () and adequately compared with a very broad spectrum of experimental observations (for a review, see Ref.s JPhysB (); Kevrekidis-et-al ().) Nevertheless, this testifies that finding exact localized solutions of the 3D GPE in a trapping external potential well, and preserving their stability for a long time, is still a challenging task. In particular, one encounters serious difficulties in attempting to find soliton solutions in one or more spatial dimensions, although several kinds of solitons have been found using certain approximations 9 (); Carr02 (). This leads us to arrive at the conclusion that, in order to have exact soliton solutions in BECs, some sort of ”control of the system” may be necessary.
The very large body of experience suggests that interacting bosons constitute a nonlinear and nonautonomous system 7 (), for which coherent stationary structures (i.e. solutions of the 3D GPE) exist only if suitable time-dependent external potentials are taken ”ad hoc” 8 (). Therefore, the correct analysis of the system should include a ’controlling potential’ term in the GPE, to be determined self-consistently with the desired solutions (e.g. the localized solutions). This procedure may be, in principle, extended to an arbitrary ’controlled solution’ with the appropriate choice of the controlling potential 10 (). The controlling potential method (CPM) has been proposed in the literature, and used to find the multi-dimensional controlled localized solutions of the GPE 10 (). Preliminary investigations 11 () have suggested that control operations introduced by this method ensures the stability of coherent solutions against relatively small errors in experimental realizations of the prescribed controlling potential. This idea could be realized by techniques that involve lithographically designed circuit patterns, providing the electromagnetic guides and microtraps for ultracold systems of atoms in BEC experiments Forthagh98 (), and by the optically induced ’exotic’ potentials Grimm00 ().
Another important aspect of solitons in BECs is the phenomenon of collective collapse/explosion, that has been predicted Carr02 () and observed experimentally Sackett99 (); Donley01 (). In particular, this phenomenon seems to be dependent on the parameters of the BECs and on the confining or repulsive potential Carr02 ().
The stabilization and control of BECs in asymmetric traps have been investigated via time-dependent solutions of the GPE Garcia98 (). Stable condensates, with the limited number of Li atoms with attractive interaction, have been observed in a magnetically trapped gas Bradley97 ().
Recently, a mathematical investigation oriented towards the construction of 3D analytical solutions of the controlled GPE has been carried out GPE1 () and applied to the construction of 3D exact localized solutions pre2009 (). In Ref. GPE1 (), it has been proven that, under the assumption of the separability of the external trapping potential well in the spatial coordinates [viz. , where and are referred to as the ’transverse’ and the ’longitudinal’ potentials, respectively], and a suitable constrained variational condition for the controlling potential (i.e. the average over the ’transverse’ plane of is required to be a stationary functional of the BEC’s transverse profile), the factorized form of the solution of the 3D controlled GPE, in the form , can be constructed, so that and satisfy a 2D linear Schrödinger equation and a 1D nonlinear Schrödinger equation, respectively.
In this paper, we apply the results of the above investigation GPE1 () to develop a new analytical procedure for constructing a broad class of exact localized solutions of the controlled 3D GPE, with a parabolic external potential well. In particular, we extend our recent investigation pre2009 () to a wider family of exact 3D localized solutions of the controlled GPE and perform an analysis that establishes the physical conditions and parameter regimes for having collapsing and non-collapsing localized solutions. In the next section, we formulate our problem and present the controlled GPE, and we briefly summarize the results found in Ref. GPE1 (). In section III, we apply these results to obtain localized solutions of the controlled GPE in the form of bright, dark and grey solitons for the longitudinal profile , and in the form of the Hermite-Gauss functions for the transverse profile . We use the decomposition procedure of the controlled GPE suggested in Ref. GPE1 () and solve the 2D transverse linear Schrödinger equation to obtain in terms of Hermite-Gauss modes. Then, the 1D longitudinal controlled NLSE is solved using a method based on the Madelung’s fluid representation Fedele02 (); FedeleSchamel02 (), which separates the NLSE into a pair of equations, composed of one continuity equation and one Korteweg-de Vries (KdV)-type equation. It is shown that the phase of the longitudinal wavefunction has a parabolic dependence on the variable . As the transverse and longitudinal equations are coupled both through the coefficient of the nonlinear term (in the longitudinal equation) and through the controlling potential, the consistency condition between the transverse and longitudinal solutions set up a relationship between the transverse and longitudinal restoring forces of the external trapping potential well. From the latter, the explicit spatio-temporal dependence of the controlling potential is self-consistently determined in each particular case. In section IV a detailed analysis of the dynamics of the above exact 3D solutions is developed both analytically and numerically. In particular, we study the properties of the controlled BEC states in the 2D case, showing that they feature breathing (oscillations of the amplitude and position) due to the oscillations of the perpendicular solution, as well as the oscillations in the parallel direction, arising from the initial displacement of the structure from the bottom of the potential well in the parallel direction. A stability analysis of controlled GPE structures is carried out in terms of a set of six parameters related through four equations. Therefore, two of them can be assumed arbitrarily. We discuss some examples of parameters corresponding to collapsing and non-collapsing 3D solutions. Finally, the conclusions are summarized in section V.
Ii Decomposition of the Controlled Gross-Pitaevskii equation
It is well known that the spatio-temporal evolution of an ultracold system of identical atoms forming a Bose Einstein condensate (BEC) in the presence of an external potential , within the mean field approximation, is governed by the 3D GPE, viz.
where is the wavefunction describing the BEC state, is the atom mass, is a coupling coefficient related to the short range scattering (s-wave) length representing the interactions between atomic particles, i.e. , and N is the number of atoms. Note that the short range scattering length can be either positive or negative. Solitons have been observed in BECs of Li atoms with a small scattering length , in correspondence of the following typical values of the parameters: – at a temperature of – and a magnetic field – r1a (); r2b ().
We assume that is the sum of a 3D trapping potential well, say , to confine the particles of a BEC, and a controlling potential, say , to be determined self-consistently. Furthermore, under this assumptions, we introduce the variable ( being the speed of light) and divide both sides of Eq.(1) by , in such a way that
and Eq. (1) can be cast into the form
where , is the Compton wavelength of the single atom of the BEC and .
where, in Cartesian coordinates, and denotes, by definition, the ’transverse’ part of the particle’s vector position and the ’longitudinal’ coordinate. Additionally, let us denote, in Cartesian coordinates, .
Let us suppose that and are two complex functions satisfying the following 2D linear Schrödinger equation
and the following 1D NLSE
respectively. In the latter, is an arbitrary real constant and the function is defined as
Hereafter is referred to as the ’controlling parameter’.
Furthermore, let us assume that the controlling potential depends in principle on through , viz. , where .
which makes stationary the functional (with respect to variation of ; and play here a role of parameters)
is a 3D solution of the controlled Gross-Pitaevskii equation (3).
In summary, according to the results given in Ref. GPE1 (), one may construct 3D solutions in the factorized form (12) such that the 3D controlled GPE is decomposed into the set of coupled equations (5) and (7) plus the self-consistent expression of the controlling potential (10) coming from a constrained variational condition. Once Eqs. (5) and (7) are solved, and become known functions. Consequently, the explicit space and time dependence of is self-consistently determined, as well, i.e. the appropriate controlling potential corresponding to the controlled solution (12) is deduced.
Note that, since satisfies Eq. (5), according to the definition (11), represents the average of in the transverse plane. The value of this average corresponds to the arbitrary constant . Without loss of generality, we may put , viz.
In this way, among all possible choices of , we adopt the one which does not change the mean energy of the system (the average of the Hamiltonian operator in Eq. (3) is the same with or without ) and therefore minimizes the effects introduced by our control operation.
In the next section, we will apply the results obtained here to the case of parabolic potentials, and , to give exact 3D controlled localized solutions of Eq. (3).
Iii Exact localized solutions of the controlled 3D GPE with a 3D parabolic potential well
Let us assume that and are the usual parabolic potential wells to confine the particle of a BEC, viz.
where, in general, the frequency , , are supposed functions of time. The standard confining potential wells (along each direction) corresponds to the assumption that they are real quantity (positivity of their squares). However, our analysis can be extended to the case in which they are assumed imaginary (negativity of their squares).
It follows that Eq. (3) becomes
According to the results and the assumptions of the previous sections, if we seek a solution in the factorized form
Eq. (16) can be decomposed into the following set of equations:
iii.1 Solution of the transverse equation with a 2D parabolic potential well
Equation (18) is readily solved, and its solution can be found in the standard literature, but we present it here for completeness. The general solution can be expressed as the superposition of Hermite-Gauss modes, viz.
where are arbitrary constants and
and the phase functions and are given by
where is an arbitrary constant.
iii.2 Exact solution of the longitudinal NLSE with a 1D parabolic trapping potential well
where is a (relatively complicated) positive definite functional of and given by
with , and
From the above equation, it is clear that, in general, depends in a non-trivial manner on and . However, in the simple case when the perpendicular solution contains only one Gauss-Hermite mode, say the -mode, Eqs. (21) and (22), becomes a real positive constant, viz.,
and, therefore, becomes
iii.2.1 Reduction of the 1D NLSE to a KdV-like equation by means of the Madelung’s fluid approach
Equation (19) is a 1D GPE with a time dependent parabolic potential. Its approximate solutions are well known in the literature, but we attempt here to find an exact solution which is also compatible with the ones of the transverse part, in such a way to give a solution of the full controlled 3D GPE (16). We seek using the following standard Madelung’s fluid representation
which, after the substitution into Eq. (19) and the separation of real and imaginary parts, yields
is the ’longitudinal potential energy’ which is a functional of .
By differentiating Eq. (32) with respect to and introducing the ’current velocity’ , a non-trivial series of transformations given in Ref.s Fedele02 (); FedeleSchamel02 () allows us to obtain the following generalized KdV equation
We look for a solution for , assuming that is a linear function of , viz.
and this corresponds to seek a quadratic form of the solution for the phase , viz.
where the ’initial phase’, , the ’wavenumber’, , and the ’dispersive’ term, , are real functions of the time-like variable . It is easy to see that the arbitrary function of , , appearing in Eq. (35), is proportional to (hereafter the prime stands for the first-order derivative with respect to ), i.e. .
where . To reduce Eq. (41) to the following KdV-like equation
we have to impose that the coefficients of and are zero, namely we automatically find that satisfies the following Riccati’s equation, viz.
while is related with it through
which is readily integrated as
where is an arbitrary constant.
We look for functions which satisfies simultaneously the KdV-like equation (42) and the continuity equation (39). By using Eqs. (31),(38) and (43)-(45), they allow us to construct also solutions of the longitudinal equation (19).
To this end, under the coordinate transformation
where the prime denotes differentiation with respect to .
To find solutions in the factorized form
Furthermore, we also observe that, given the set of Eqs. (43), (45), (53) - (55), we can conveniently express the functions , and in terms of the controlling parameter (which is directly connected with the transverse part of the GPE solution), i.e.
In particular, Eq. (59) has been obtained by substituting the first of equations (50) into Riccati’s equation (43). It establishes a ’control condition’ by the transverse part of the GPE solution on the longitudinal parabolic potential. In fact, it indicates which time dependence of has to be taken, provided that , given by Eq. (26), is solution of the first of Eqs. (50). Note that, according to Eq. (26), it results that
(Note that does not coincides with .) To cast Eq. (52) as an equation with constant coefficients, we can choose the arbitrary function proportional to and therefore
where is an arbitrary constant, in such a way that, according to Eq. (45) and the definition of given above, we have
Let us now determine the phase . To this end, we first observe that can be obtained by integrating Eq. (61), i.e.
iii.2.2 Soliton solutions
As it is well known, Eq. (63) admits both localized and periodic solutions Sulem99 (); Whitham99 (); Dau06 (). Typically, the latter are expressed in terms of Jacobian elliptic functions, whose suitable asymptotic limits of their parameters recover the localized solutions in the form of bright, dark and grey solitons. However, a very useful integration approach of Eq. (63) that has a simple physical meaning of the solutions is the well-known Sagdeev’s pseudo-potential method Kadomtsev (); Karpman ().
(i). Bright solitons
If (and consequently ) is a normalizable wave function, we can look for a bright soliton solution of Eq. (63), which satisfies the following boundary conditions: , for . This solution exists for , FedeleSchamel02 (), i.e.
The positivity of implies that and therefore and . On the other hand, if we require that is normalized, i.e.
It follows that, in the case of bright solitons, Eq. (67) becomes
(ii). Grey and dark solitons
If (and consequently ) is a non-normalizable solution, we can look for dark or grey soliton solutions of Eq. (63), which satisfy the following boundary conditions: , for , where . These solutions are given by the general form
where is a real parameter,
We first observe that since , the following condition holds:
which, going back to the variables and , can be cast into the form