Department of Mathematics and Statistics, University of Massachusetts,
Amherst MA 01003-4515, USA

R. Carretero-González,
Nonlinear Dynamical System Group (NLDS), Computational Science Research Center, and Department of Mathematics and Statistics, San Diego State University,
San Diego, California 92182-7720, USA

D.J. Frantzeskakis,
Department of Physics, University of Athens, Panepistimiopolis, Zografos,
Athens 157 84, Greece

Abstract: In this brief exposition, we showcase some recent experimental and theoretical work in the coldest temperatures in the universe involving topological defects (vortices) in the newest state of matter: the atomic Bose-Einstein condensates. The remarkable feature that these experiments and associated analysis illustrate is the existence of a new kind of ``classical mechanics'' for vortices, which revisits the integrability of the two-body (i.e., two-vortex) system and opens up exciting extensions for N-body generalizations thereof. One-dimensional and three-dimensional analogues of such dynamics, involving dark solitons and vortex rings, respectively, are also briefly touched upon.

The General Setting: Bose-Einstein Condensates and Vortex Emergence

The phenomenon of Bose-Einstein condensation [1,2,3] is a quantum phase transition originally predicted in 1924. In particular, it was shown that below a critical transition temperature T_c a macroscopic fraction of particles of a boson gas (i.e., whose particles obey the Bose statistics) condenses into the same quantum state, known as the Bose-Einstein condensate (BEC). Despite the fundamental nature of this prediction, BECs were not experimentally realized until 70 years later: this major achievement took place in 1995 and has already been recognized through the 2001 Nobel Prize in Physics [4]. This first unambiguous manifestation of a macroscopic quantum state in a many-body system sparked an explosion of activity, as reflected by the publication of several thousand papers related to BECs since then. Nowadays there exist more than fifty experimental BEC groups around the world, while an enormous amount of theoretical work has followed and driven the experimental efforts, with an impressive impact on many branches of Physics.

From a theoretical standpoint, and for experimentally relevant conditions, the static and dynamical properties of a BEC can be described by means of an effective mean-field model, namely a partial differential equation (PDE) known as the Gross-Pitaevskii equation (GPE). This is actually a nonlinear Schrödinger (NLS) equation [5] of the form:

where Ψ = Ψ(r,t) is the macroscopic wavefunction of the condensate (the atomic density is proportional to |Ψ(r,t)|²), Δ is the Laplacian, m is the atomic mass, and the prefactor g is proportional to the atomic scattering length and describes the interatomic interactions [6,7] (the latter may be repulsive, \(g > 0\), e.g., for ⁸⁷Rb and ²³Na atoms, or attractive, g < 0, e.g., for ⁸⁵Rb or ⁷Li atoms). The external potential V_ext is used to confine the atoms and, usually, has a harmonic (parabolic) form.

One of the remarkable features of this GPE-based mean-field approach is that it allows the prediction and description of coherent macroscopic excitations that can be supported in BECs due to the interatomic-interaction-induced nonlinearity in Eq. (1). Relevant ``nonlinear matter-waves'' include bright, dark and gap solitons, as well as vortices and vortex lattices [6,7]. The present discussion will especially focus on vortices whose profound implications in fundamental physical phenomena, such as superconductivity and superfluidity, were also recognized by the Nobel Prize in Physics in 2003 [8].

Vortices are ubiquitous circulating flow patterns that arise in numerous contexts [9], ranging from hydrodynamics, superfluids, and nonlinear optics to specific realizations in sunspots, dust devils [10], and plant propulsion [11]. In BECs, quantized vortices arise as persistent topological defects that have a key role in both Hamiltonian and dissipative dynamics, as well as in quantum turbulence. The relevance of such individual coherent topological structures, as well as of large scale lattices thereof, has been analyzed in a series of specialized reviews [12,13,14]. It should be noted here that while vortices can arise in BECs with attractive inter-atomic interactions (g < 0), their instability towards catastrophic collapse-type events renders them less robust therein. We will thus, hereafter, focus on condensates with repulsive interactions (g > 0), where vortices are very robust.

A Special Case of Particular Interest: The Vortex Dipole

In the case of repulsive ``pancake''-shaped BECs confined in strongly-anisotropic (quasi-two-dimensional) harmonic traps, a single vortex has relatively simple dynamics: if placed at the center of the trap, it can be identified as a stationary mode of the nonlinear system of Eq. (1). On the other hand, if placed off-center, it does what a (topologically) charged particle does in a magnetic field: it precesses with frequency Ω. Matched asymptotics [12] can be used to obtain this precession frequency near the center of the trap, as well as its logarithmic dependence on the trap strength and on the chemical potential (the latter, pertains to the strength of the nonlinearity or, physically, the number of atoms of the BEC). Figure 1 shows an example of such a stationary vortex and of its precessional space-time (off-center) dynamics. Notice that this bears a direct analogy to the simple oscillatory dynamics of a single dark soliton (a one-dimensional density dip) in quasi-one-dimensional, ``cigar''-shaped BECs [15].

Figure 1: Prototypical example of the a) amplitude and b) phase profiles of a trapped vortex at the center of a parabolic trap. c) (x,y,t) dynamics for an off-center precessing vortex from the full GPE (1) (blue points) and the reduced ODE model (2) (thin black line). d) Animation of the evolution of the density for a precessing vortex.

Things get far more interesting when considering the evolution of two (or more) vortices. Then, each vortex has its own precessional dynamics, but importantly there also exists the pairwise vortex interaction which can be thought of as a Hamiltonian fluid point vortex interaction [16]. The resulting (reduced) dynamical set of ordinary differential equations (ODEs) of motion characterizing the center positions (x_k,y_k) of N interacting vortices reads:

where z_k = x_k + iy_k = r_k e^iθ_k, r_k = |z_k| and r_jk = |z_k-z_j|. In Eq. (2) the topological charge of the k-th vortex is S_k = ± 1, with the positive (negative) sign referring to counterclockwise (clockwise) circulation as viewed from the positive z axis. Also, the precession frequency that can be accurately approximated by \(\Omega(r_k) = \Omega_0/(1-r_k^2/R^2)\) (for \(r_k < R\)) depends on the spatial location of the vortex [12,14], where R denotes the spatial extent of the BEC and Ω₀ is the (spatially independent) precession frequency value at the center of the BEC, while A is a constant prefactor that accounts for the spatial inhomogeneity of the BEC [17].

While the above equations describe an arbitrary number of trapped vortices, the case of two opposite charge (i.e., counter-rotating) vortices, i.e., a vortex dipole, is especially motivated by a series of recent experiments. In these experiments, dipoles were produced either by dragging an obstacle through the BEC [18] (a superfluid analog of the classic flow-past-a-cylinder experiment), or by quenching rapidly through the quantum phase transition (trapping phase singularities in the process) [19,20] in what is known as the Kibble-Zurek mechanism. The case of two vortices is especially interesting as it forms a four-degree-of-freedom Hamiltonian system, in which -in addition to the standard conservation law of the energy- also the angular momentum is conserved [21]. The presence of two conserved quantities or dynamical invariants guarantees integrability in the classical Liouville sense for the case N=2, whether the vortices are co- or counter-rotating. This implies that the energy level sets are compact and the phase space is foliated by invariant tori. On each of these, the motion is generically quasi-periodic with two frequencies. In the following, we consider further dynamical aspects of the N=2 case for vortex dipoles with S₁ = - S₂ = 1.

In particular, since the precession tends to rotate each vortex in one direction, while their mutual interaction yields an opposite tendency, an equilibrium distance of the dipole can be identified in excellent agreement between the above particle ODE picture (2) and the experiment [20]. In some sense, this ``zero frequency'' equilibrium state is the simplest (non-generic) dynamical scenario among the possible vortex dipole motions. The immediately next (still non-generic) scenario in terms of complexity pertains to a ``single frequency'' motion: such an internal mode of the two-vortex system can also be identified by linearizing around the above equilibrium. It corresponds to an epicyclic precession of the vortices around the equilibrium that was also observed in experiments (see Fig. 2) again in good agreement with the theoretical predictions [20].

Figure 2: Panel a) depicts experimental data snapshots for the epicyclic precession of two vortices as a function of time (indicated by the different colors). The data are assembled into a trajectory in panel b): the experimental vortex locations are given by triangles, while the theoretical predictions for the same times are shown by circles (of the same color). The underlying theoretical trajectory is given by the solid line. Adapted from Ref. [20].

Finally, the typical scenario in this dipole setting concerns quasi-periodic (generic) motions. Namely, when two frequencies are involved in the motion of each vortex, we see the more complicated either ``epitrochoidal'' motions of the top panel, or the outer fast precession and inner flower-like motions of the middle and bottom panels of Fig. 3. All of these orbits, however, in the proper co-rotating frame result in closed orbits revealing the quasi-periodic character of the motion. In fact, although the particle dynamics is integrable, the underlying full experimental system to which it is compared can be thought of as a weakly non-integrable (infinite degrees of freedom) generalization thereof. As such, the observed identification of quasi-periodic orbits can be thought of as an experimental manifestation of KAM theory. It should also be mentioned in passing that this class of particle models parallels the understanding now reached in the case of the oscillations and interactions of the one-dimensional analogues of vortices, namely the dark solitons [15]. There, a model in the form of a Toda lattice on the relative positions between adjacent dark solitons (dark soliton interactions) embedded in a harmonic trap (effect from the parabolic trapping on each dark soliton):

can be used to characterize the orbits of the interacting dark solitons as a function of system parameters (such as the effective strength of the parabolic trap ω_trap and chemical potential μ).

Figure 3: Same as in the previous figure for 3 different trajectories a), b), c) which are generic examples of the two-frequency dynamics of the vortices. Panels d), e), f) summarize the vortex motions and illustrate (together with a), b) and c)) the agreement between ODEs and experiment. Panels g), h) and i) explicitly showcase the quasi-periodic nature of these motions, by virtue of displaying the dynamics in a proper co-rotating frame (whose rotational frequency is indicated in each panel). Adapted from Ref. [20].

Generalizations and (Many) Open Avenues

Although the vortex dipole is gradually becoming a more charted territory and provides the fundamental modeling background for tackling more complex settings, in practice there is a tremendous amount of open problems that are shaping up within this field, as stemming from its novel ``classical mechanics'' of these interacting vortex particles (or, for that matter, the interacting dark soliton particles). Perhaps the simplest such issue consists of the co-rotating vortex pair which, although integrable, has been recently shown to possess an instability of its prototypical configuration of two equi-distantly rotating vortices (with respect to the center of the trap). An interesting symmetry-breaking bifurcation already seems to arise therein.

Going beyond N=2 vortex (or soliton) dynamics breaks the integrability and poses exciting challenges in its own right. Are there chaotic trajectories, (analogues of) Lagrangian points and other interesting features reminiscent of classical 3-body problems of classical mechanics ? It should be noted here that 3-vortex states have also been identified in experiments in either co-rotating or counter-rotating sets of clusters in the very recent experiments of Ref. [22], hence there is a significant interest towards their detailed understanding. Generalizing notions to N-body problems, e.g., identifying special equilibrium configurations, examining their stability, and the near equilibrium (or far from equilibrium) dynamics, as well as studying possible periodic orbits tantamount to the periodic choreographies (cf., figure-eight N-planet orbits) found in celestial mechanics [23,24], constitute further exciting frontiers in these explorations. An additional frontier concerns the transitions of this system from the near-equilibrium soliton/vortex-solid to the far-from-equilibrium soliton/vortex gas and the potentially turbulent dynamics that the latter may possess. Clearly this dynamics is beyond the above particle picture and should be considered at the level of the full PDE of Eq. (1) and, most likely, of its dissipative generalizations pertinent to finite temperature BECs. Such issues are starting to be explored in both the theoretical and experimental literature of BECs and may produce an exciting framework for exploring, among others, superfluid quantum turbulence.

Figure 4: Vortex ring interactions, mergers and breakups. The left animation depicts two vortex rings chasing each other in a ``leap-frog'' orbit. The middle animation depicts four co-planar vortex rings that attract each other, merge and in the process shoot a fifth vortex ring in the opposite direction. The right animation depict the periodic merger and breakup of five co-planar vortex rings.

Finally, while two-dimensions already afford the possibility for topological charges, the fully three-dimensional setting enables the consideration of far more complex fundamental entities than the line-vortex generalizations of our two-dimensional vortices above, namely the study of vortex rings. These coherent structures also possess fascinating dynamics including leap-frogging orbits between two vortex rings and remarkable merger and breakup events in the case of more rings (see Fig. 4 for some animated examples). Developing a particle picture and analyzing the full three-dimensional PDE evolution pose exciting tasks not only for experiments and theory, but even for numerical implementations thereof.

Acknowledgments

We would like to acknowledge our colleagues D.S. Hall, D.V. Freilich, S. Middelkamp, P.J. Torres, P. Schmelcher, and R. Navarro for their contribution towards this ongoing collaboration. We also thank R.M. Caplan for the vortex ring simulations and NSF-DMS-0806762 and the Special Account for Research Grants of the University of Athens for financial support.

References