A quasi-local Gross–Pitaevskii equation for attractive Bose–Einstein condensates
Introduction
One of the basic model equations in the field of nonlinear waves with a wider range of applications, is the nonlinear Schrödinger equation (NLSE). It arises, in different forms, in nonlinear optics, polymer-molecule dynamics, plasma physics, many-body quantum systems treated in the Hartree approximation, Bose–Einstein condensation (BEC), and many other problems [1].
A little explored type of NLSE is which includes a nonlocal kernel and an external potential . In the case of the local kernel, , Eq. (1) goes over into the usual NLSE with an external potential.
Nonlocal models of this type appear in different contexts: nonrelativistic baryon models, quantum gravity [2], fundamentals of quantum mechanics, thermal self-focusing of laser beams, liquid-helium BEC, and, more recently, BEC with ultracold atomic gases [3], [4], [5], [6] and other problems. On the mathematical side, most of the work has concentrated on a specific instance of Eq. (1) which is usually referred to as the Schrödinger–Poisson [7], [8] or Schrödinger–Newton [2] system.
It is known that when a weakly interacting boson gas with dominating two-body interactions is cooled down below a certain temperature, it undergoes a phase transition to a collective state which is called a BEC. The theoretical description of such a BEC is provided by the so called Gross–Pitaevskii equation (GPE) [9]where m is the boson mass, the trapping potential, and U0=4πℏ2a/m is proportional to the scattering length a and characterizes the two-body interaction. The case a>0, corresponds to repulsion between the bosons. In this situation solutions to Eq. (2) are well-defined for all times. However, in the case of attractive interactions, a<0, collapse (blow-up) may take place in the 2D and 3D cases. The presence of the collapse means that there is a generic set of initial conditions such that the solution ceases to exist after a finite of time. The evolution of collapsing solutions is a fundamental point in the theory of multidimensional nonlinear Schrödinger (NLS) equations [10]. Many results on collapse in BEC systems (i.e. including the trap) have been obtained previously both in the theoretical (see [11], [12], [13] and references therein) and experimental sides (see [14] and references therein).
As it has been mentioned above, Eq. (2) is a local model which approximates the more realistic case of nonlocal interactions in which the potential of the two-body interaction between bosons is given by: The nonlocal kernel contains the details of the interaction. The Hamiltonian then reads so that the wave function satisfies a nonlocal NLSE which is [15]In accordance with the common situation in BEC we choose the trapping potential to be , where ν is a characteristic frequency, and (α=x,y,z) describe the asymmetry of the trapping potential. However, many of our results are also applicable to the case when λj=0, as it will be discussed later.
For positive scattering lengths the nonlocality of the interactions manifests itself only as small corrections [5], but in the case of attractive interactions the nonlocality of the potential is essential. For particularly simple kernels it has been proved in Ref. [16] that the Hamiltonian is bounded below, which is the first step towards a full proof of the global existence of solutions. Qualitative arguments in favor of preventing collapse by nonlocal interactions were also proposed in Ref. [3]. Besides that, in Ref. [4] it has been argued and shown numerically that nonlocality gives rise to oscillations of a wave-packet between the large scale defined by the trap potential V and a small scale determined by the interaction radius. Indeed, it has been found [4] that at the point when the development of collapse stops, the wave-packet width is still significantly larger than the intrinsic size of the kernel K (3). This raises the question whether it is important to keep the nonlocality in its exact form, or it can be modeled by a simplified quasi-local model. Another reasons for the interest of this type of models will be given later.
It is our intention in this paper to provide a general and simple description of the nonlocal interaction using a quasi-local model which retains essentials of the dynamics of the problem.
Section snippets
The quasi-local model
We first transform Eq. (5) into a renormalized form by defining new variables, , . Since we will only be using the rescaled variables, we omit the tildes attached to them, so that accordingly transformed Eq. (5) takes the form Note that the normalization of the rescaled wave function gives the number of particles in a given BEC state, .
The standard
Self-similar solutions
The simplest version of Eq. (10) is that with the radial symmetry. Radially symmetric solutions are interesting since they allow direct insight into the collapse process [10], [18]. Thus we will look for solutions in two (D=2) or three (D=3) dimensions. In this case, Eq. (10) reads To analyze the existence of collapsing solutions it is customary to renormalize the radially symmetric solution with a scale factor L(τ) [18]
Exact results
While the variational approach is intuitively appealing, one cannot make any definite statements based on this approximation. Also, up to now, our analysis has been restricted to solutions with radial symmetry. In this section we present more rigorous arguments supporting the idea that collapse is prevented in Eq. (10). We note that it was previously argued in Ref. [22] that a quasi-local model of the present type is collapse-free in 2D. Here we present a more rigorous proof of this assertions
Solitary waves: ground states
Let us consider solitary wave solutions to Eq. (10) in the form These solutions are critical points of the Hamiltonian given by Eq. (4). The fact that they are critical points means that the variational derivative of H[ψ], for a fixed norm, vanishes, i.e. δH/δψ|∥ψ∥22=N=0.
It is interesting to characterize these solutions since they correspond to stationary configurations of BEC with the nonlocal attractive self-interaction. In particular we will be interested in the solution
Conclusions and discussion
In this paper we have proposed a quasi-local model with a nonlinear dispersion term which approximates the nonlocal term describing interactions between bosons in a Bose–Einstein condensate. Our quasi-local description retains all the essential features of the full nonlocal model: collapse prevention, dependence of the width of the ground state on the interaction range, oscillatory behavior, and insensitivity of the eventually established soliton-like state to the particular trapping potential.
Acknowledgements
JJG-R and VMP-G have been supported by CICYT under grant BFM2000-0521. The cooperative work has been supported through the bilateral program DGCYT-HP1999-019/Ação no. E-89/00.
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