Original ArticlesExcited Bose–Einstein condensates: Quadrupole oscillations and dark solitons
Introduction
The realization of Bose–Einstein condensates (BECs) in dilute atomic vapors, which was awarded the Nobel prize in physics in 2001 [9], [15], has been a fundamental development in quantum and atomic physics in the last decades. The statics and dynamics of BECs can be described by means of an effective mean-field model, known as the Gross–Pitaevskii (GP) equation [10], which is a variant of the well-known nonlinear Schrödinger (NLS) equation. This description allows for an understanding of fundamental properties of BECs, including their collective excitations [10]. These refer to the internal oscillatory modes of the system and confirm the superfluid nature of the condensate. They are usually studied in the framework of the hydrodynamic approach for BECs, but also in the context of the ensuing Bogoliubov–de Gennes (BdG) equations when linearizing around its stationary states [10]. Importantly, collective oscillations have also been studied experimentally, with a pertinent example being the so-called quadrupole oscillation [20], [27]. This oscillation was observed in an axisymmetric cigar-shaped BEC, which executed shape oscillations due to a temporal variation of the trap frequency. The value of the characteristic frequency of the quadrupole oscillation found in the experiment was in excellent agreement with the one predicted theoretically [1], [23], [28].
Apart from collective oscillations, the GP mean-field approach is able to describe a variety of purely nonlinear macroscopic waveforms, such as solitons and vortices [8], [16]. Among the most fundamental nonlinear excitations that are supported in BECs with repulsive interatomic interactions, are the so-called matter-wave dark solitons, which have received much attention (see the recent reviews [8], [17] and references therein) due to their experimental realization [2], [3], [5], [7], [11], [12], [14], [24], [26], [30], [32]. The basic properties of dark solitons, such as their statics, stability and dynamics, have mainly been investigated – in the quasi one-dimensional (1D) setup of cigar-shaped BECs – in the case of a static background. More recently, different approaches have been used to study the dynamics of dark solitons in various time-dependent traps, which render the background time-dependent as well [4], [25], [33].
In this work, we study the dynamics of excited BECs, namely breathing condensates, which may additionally carry a dark soliton. Employing the so-called Thomas–Fermi (TF) approximation [10], we introduce an ansatz for the BEC wave function: this has the form of a chirped TF cloud and includes unknown time-dependent functions that describe the evolution of the density and phase of the breathing BEC. This way, we obtain a set of ordinary differential equations (ODEs), which is solved analytically for two different cases: when quadrupole oscillations of the condensate occur due to an abrupt change of the trap strength, or when breathing oscillations of the condensate arise due to a time-dependent trap strength. Our analytical approximations rely on the assumption that, in either case, the change of the trap strength that induces the BEC oscillations is small. Then, this change is used as a formal small parameter in a perturbation scheme which results in a very accurate description of the breathing BEC dynamics. Thus, the analytical results are found to be in very good agreement with direct numerical simulations of the GP model.
Having determined the evolution of the BEC density and phase, we also investigate numerically the evolution of a dark soliton on top of such oscillating backgrounds. It is found that for sufficiently weak background oscillations, the soliton dynamics resembles the one occurring in a static BEC: the soliton oscillation is roughly harmonic, with a frequency close to the characteristic value , where Ω is the harmonic trap strength [4], [6], [13], [18], [22], [29]. Nevertheless, in the case of large changes of the trap strength, the motion of the dark soliton becomes more complicated; even in such a case, however, the systematic analysis of the oscillations of the breathing background allows for a qualitative understanding of the spectrum of the soliton trajectory.
The paper is organized as follows. In Section 2 we present our model and study analytically the quadrupole oscillation of the background wave function induced by an abrupt change of the trap strength. In Section 3 we extend the previous analysis to a case where breathing oscillations of the condensate are induced by a time-dependent trap strength. Section 4 presents numerical results for the dynamics of dark solitons in breathing BECs and, finally, in Section 5 we present our conclusions.
Section snippets
General background
We consider a quasi-1D repulsive BEC lying along the x-direction, which is confined in a highly anisotropic trap with frequencies ωx ≪ ω⊥. For sufficiently low-temperatures, the macroscopic wave function ψ(x, t) of the condensate is described by the following effectively-1D GP equation [8], [16]:where is the external confining potential in the longitudinal direction, and g1D = 2αℏω⊥ is the effective 1D coupling constant. Measuring density, length,
Oscillations of the condensate due to an abrupt change of the trap strength
Let us now assume that the quadrupole mode is excited by means of a small change of the trap strength. In particular, we assume that, at t = 0, the trap strength undergoes a step-like change, from the value to the value Ω, where the formal small parameter ϵ (chosen so that ϵ / Ω ≪ 1) defines the change of the trap strength.
In such a case, a solution of Eq. (18) can be found by exploiting the smallness of the change in the trap strength. This way, we first introduce the expansion:
Oscillations of the condensate in a time-dependent trap
Let us now consider the excitation of the quadrupole mode by applying a time-dependent trapping potential. In particular, we assume that the trap strength takes the following form,where, Ω0 is the trap strength (in the absence of the harmonic driving term), while the parameters ϵ and ν denote, respectively, the strength and the frequency of the driver. Assuming that ϵ is a formal small parameter, we look for analytical solutions of Eq. (18) using the expansion in Eq. (26).
Dark solitons in breathing condensates
Let us now consider the case where a condensate performing quadrupole oscillations also carries a matter-wave dark soliton. We assume that the breathing oscillations of the BEC are excited by either of the two different methods analyzed in the previous sections; furthermore, in our simulations (see below), we will assume parameter values similar to the ones used above, so as to facilitate a direct comparison of the cases where the dark soliton is absent or present on top of the breathing BEC.
Conclusions and discussion
In summary, we studied the dynamics of quasi-1D atomic BECs that perform breathing oscillations. We considered two different situations: (i) the quadrupole mode of the BEC was excited by means of a small abrupt change in the trap strength and (ii) the trap strength was assumed to be time-dependent (with a small-amplitude harmonic change around a constant value). Both situations can be straightforwardly realized experimentally.
Assuming in either case that the change in the trap strength is
Acknowledgments
The work of D.J.F. was partially supported by the Special Account for Research Grants of the University of Athens. P.G.K. gratefully acknowledges support from NSF-DMS-0806762, NSF-DMS-0349023 and the Alexander von Humboldt Foundation. We also thank the anonymous referees for many helpful suggestions.
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